CN105046079A - D-optimal inner table design based Taguchi experimental design method - Google Patents

Abstract

The present invention provides a D-optimal inner table design based Taguchi experimental design method. The method comprises specific steps of: 1. providing basic experimental information including experimental responses, factors, inter-factor restrictions and the like according to experimental objectives, conditions and engineering experience; 2. selecting a proper uniform design table according to the number of noise factors and the number of proposed levels; 3. determining a relationship model between the responses and the factors, and the number of times of inner table experiments; 4. in a design area, providing an inner table design scheme with given experimental times by using the D-optimal design method, and completing the inner table design; 5. completing external table design by using an empirical distribution function based point equal-division method; 6. performing an experiment according to an experimental scheme, calculating a signal-to-noise ratio of mass properties by using performance indexes obtained from the experiment, and calculating the optimal level of a controllable factor that enables the signal-to-noise ratio to be maximized; 7. determining the optimal level of a stable factor by a sensitivity analysis method; and 8. changing and adjusting the factors to determine an optimal factor level combination.

Description

A kind of field mouth test design method based on showing in D-optimum to design

Technical field

The present invention relates to a kind of field mouth test design method based on table design (namely to change greatly the method for designing that testing site determinant (Determinant) is criterion) in D-optimum, not only effectively can solve traditional field mouth and design the irregular situation of reluctant design section, and effectively can alleviate the problem that mouth design in field adopts direct product sheet form to cause test design number of times to increase severely, be applicable to the correlative technology fields such as product design, the manufacturing, quality control and process optimization.

Background technology

A rational test design scheme can under the resources supplIess such as limited time, cost, each factor of influence of abundant announcement is on the impact of target response, under limited test number (TN), obtaining optimum target response result, seeking optimum combination of process parameters etc. for improving the quality of products.

In order to the stability of enhancing product performance, a mouth profound doctor in field proposes field mouth method for designing, the form of table, appearance direct product in adopting, each level of controllable factor is arranged in interior table, in appearance, arrange noise factor to test, and so-called direct product method, refer to each collocation of controllable factor in internally showing, the error factor of appearance is used to simulate various interference, calculate the antijamming capability of this collocation, i.e. signal to noise ratio (S/N ratio) (SN ratio), thus by internally showing the comparison of the various scheme of design, utilizing SN ratio to find best controllable factor and arranging in pairs or groups.

Inventionbroadly, all test design itself are all resource-constrained designs, and the object of test design is exactly when practicing every conceivable frugality resource, obtain as far as possible many systems or information, optimal design and the technique of process.In traditional field mouth design, interior table, appearance all use orthogonal design to carry out, and carry out data analysis by variance analysis, and its design section requires it is regular hypercube, generally do not consider there is constraint between the factor and cause the irregular situation of design section.In Practical Project, owing to being often not independently between factor of influence, there is certain interaction, thus produce the irregular situation of design section, at this moment, use orthogonal design to carry out the design of interior table, need cut or fill up compromise, contrast that pilot region carries out repeatedly, not only design process complexity, and due to dimensionality reduction too much, cause design efficiency lower.Meanwhile, the direct product sheet form of field mouth design, makes when noise factor is more, and test number (TN) increases severely, and often exceeds the scope that test resource allows, causes enterprise to be difficult to bear; In addition, because in traditional field mouth, appearance design uses orthogonal design to carry out, and orthogonal design is the design based on design table, cause field mouth design experiment number of times not adjust flexibly according to the actual requirements, easily cause resource not make full use of or the situation of inadequate resource.Therefore, study when factor of influence exists constraint, reasonably Selection experiment point, thus scientifically Collection and analysis data, obtain more excellent parameter combinations, optimal design and technique, have important theory significance and urgent current demand.

For this reason, The present invention gives a kind of field mouth test design method based on showing in D-optimum to design.

Summary of the invention

(1) object of the present invention: the present invention is directed to traditional field mouth design in limited test number (TN), be difficult to flexible and efficient solution pilot region irregular time test design problem, a kind of field mouth test design method based on showing in D-optimum to design is provided, have more representational testing site to filter out, thus carry out accurately, efficiently, plan design flexibly.

D-optimal design is a kind of method for designing based on mathematical model, according to model parameter number, in given design section, can provide the optimal design meeting any test number (TN) within the scope of certain condition; Under the framework that the present invention designs at field mouth, D-optimal design is utilized to carry out the design of interior table, when can there is constraint when between the factor, provide more flexible, efficient test design scheme, set up the mathematical model of more pressing close to Product processing or production run reality, carry out data analysis more accurately and prediction, thus determine best factor level combination.

(2) technical scheme:

The present invention gives a kind of field mouth test design method based on showing in D-optimum to design.

The method that field mouth design is shown in using and appearance combines carries out based Robust Design, and uses signal to noise ratio (S/N ratio) (SN) as quality evaluation index, to seek the most stable parameter combinations.

Traditional field mouth design, interior table adopts orthogonal design to arrange the test of controllable factor, and appearance adopts orthogonal design to arrange the test of noise factor, uses the size of signal to noise ratio (S/N ratio) (SN) Measure Indexes fluctuation.Due to the limitation of orthogonal design, when design section is irregular, traditional field mouth method for designing cannot provide efficient design proposal.In addition, orthogonal test number of times is determined by method for designing, but not test accuracy, under the effect of field mouth design direct product table, the character of orthogonal design " neatly comparable " makes test number (TN) increase severely, and adjustment is very dumb.But, in snr computation process, do not require each error component in testing program " neatly comparable ", the requirement of " neatly comparable " therefore, can be loosened, explore more efficiently test design method.

The present invention adopts D-optimal design to carry out the design of interior table, adopts the uniform Design based on quantiles such as empirical distribution functions to carry out appearance design.By general equivalence theorem, for the linear model having p parameter, exist by n ^*(p≤n ^*≤ p (p+1)/2) the D-optimal design ξ of individual testing site composition ^*.Namely D-optimal design with within the specific limits, can provide the interior watch test design proposal of any design section, arbitrarily test number (TN), and along with the increase of test factor number, the minimum test number (TN) required for D-optimal design will be far smaller than other designs.Under the prerequisite meeting test accuracy, D-optimal design is used to carry out the design of interior table, and carry out appearance design in conjunction with uniform Design, not only effectively can solve traditional field mouth and design the situation that there is constraint between the reluctant factor, and under less test number (TN), efficient test design scheme can be provided.

Based on above-mentioned theory and thinking, a kind of field mouth test design method based on showing in D-optimum to design of the present invention, concrete implementation step is as follows:

Step one: according to test objective, condition and engineering experience, provide test fundamental test information, comprise test response (being paid close attention to mass property), factor of influence (comprising controllable factor and noise factor) and span thereof, the test total degree n that the restriction relation between each factor and test resource allow ^total.

Step 2: the horizontal number m adopted according to number and the plan of noise factor, selects suitable uniform designs table, and determines appearance test number (TN) n with this ^outer.System of selection brief introduction is as follows:

Suppose there be noise factor l, intend adopting number of levels m, then from uniform designs table storehouse, select evenly table appearance test number (TN) is defined as n simultaneously ^outer=ω.

Step 3: determine to respond the relational model between the factor and interior watch test frequency n ^inner.

The concrete defining method of relational model in this step between response and the factor is as follows:

Based on the test essential information in step one, adopt the form of mathematical model to give expression to the relational model responded between the factor, be designated as y=X β+ε.Wherein, for known n ^inner× p ties up parameter model matrix; F (x _i) be about x _iknown function, react whole controllable factor and between restriction relation; β=(β ₀, β ₁, β ₂..., β _p) ^tfor p solve for parameter, react the interact relation between each factor and response.

Interior watch test frequency n ^innerdefining method as follows:

By general equivalence theorem, for the linear model having p parameter, exist by n ^*(p≤n ^*≤ p (p+1)/2) the D-optimal design ξ of individual testing site composition ^*.So, according to overall test frequency n ^totalwith appearance test number (TN) n ^outer, be not difficult to provide [p, the min (p (p+1)/2, n of watch test numbers range in D-optimum ^total/ n ^outer)].

Therefore, according to actual test conditions, select any n ^inner∈ [p, min (p (p+1)/2, n ^total/ n ^outer)].

Step 4: in design section, adopts D-optimal-design method to provide given test number (TN) n ^innerinterior table design proposal, complete interior table design, its specific design method is as follows:

First, by each factor span specification in [-1,1] interval range, according to the restriction relation between the actual factor, design section is determined; Then, in this design section, select the design ξ meeting given number of times, make information matrix determinant is maximum, so far completes table design in D-optimum.Further, G-optimum efficiency is used to weigh the Optimality of this design.G-optimum efficiency is defined as:

$G_{eff}=\frac{p}{\underset{x\∈X}{max}d(x,\mathrm{\ξ})}$

Here, G-optimum efficiency value G _effmore close to 1, represent that design ξ is better.

Step 5: adopt based on quantile methods such as empirical distribution functions, for appearance testing site each in interior table, determines noise factor level, completes appearance design.

In this step, the defining method of noise factor level is as follows:

Note x _[i]for variable x based on quantile such as empirical distribution function i-th grade, it is defined as follows:

$x_{\[i\]}=\arg F\left(\frac{i}{m}\right),i=1,2,...,m$

Wherein, the distribution function that F (x) is stochastic variable x, m is noise factor number of levels; The inverse function of argF () representative function F ().In other words, above formula represents, when time, the value of corresponding stochastic variable x is designated as x _[i], x _[i]be i-th level value of factor x.

Step 6: specifically test according to testing program, according to testing the product performance index calculated mass characteristic signal to noise ratio (S/N ratio) obtained, matching obtains the regression function η (x between signal to noise ratio (S/N ratio) and the factor _i), calculate and make the maximized controllable factor optimum level of signal to noise ratio (S/N ratio).

Step 7: based on Sensitivity Analysis Method determination stable factor and optimum level thereof.

Signal to noise ratio (S/N ratio) is to factor of influence x _i, i=1,2 ..., the sensitivity definition of n is as follows:

S _ijnumeric representation to variable x _isensitivity, S _ijsymbol represent to variable x _imonotonicity.The S that numerical value is larger _ijthe corresponding factor is stable factor, and get the combination of its optimal level with the stability reaching system, the fluctuation that system is exported is reduced to a minimum.Its complementary divisor is Dynamic gene, and the output valve of adjustment System makes it to reach or close to desired value, thus when reducing fluctuation as far as possible, obtains mass property is combined closest to the optimal parameter of its desired value.

Step 8: regression fit tries to achieve unknown parameter in relational model y, and adjusts Dynamic gene according to this, determines optimum factor horizontal combination, to make under this testing program mass property estimated value closest to desired value.

(3) advantage and effect:

The invention provides a kind of field mouth test design method based on showing in D-optimum to design, its advantage is:

1. this utilizes D-optimal design to carry out the design of interior table, between test factor under Existence restraint condition, can provide efficient and rational field oral examination and test design proposal.

2. the present invention effectively alleviates the test number (TN) sharp increase problem that field mouth design direct product sheet form causes, and interior table can provide the test meeting arbitrary number of times under certain condition, and testing program number of times can be adjusted flexibly.

Accompanying drawing explanation

Fig. 1 is the inventive method process flow diagram.

Embodiment

Below for bead nitridation process, by reference to the accompanying drawings, the present invention is described in further details.

Feedback spring is the critical component of electrohydraulic servo valve, and the bead on its feedback rod and the feedback groove on spool require to be in zero stand-off mated condition.In the use procedure of feedback spring, if bead hardness surface is not, easily causes wear phenomenon, cause no load discharge curve to suddenly change, thus the servosignal that leads to errors, make electrohydraulic servo valve produce critical fault.In order to improve bead surface hardness, need to carry out ion soft-nitriding process to bead.Nitride layer depth is darker, then the wearing quality of bead is better, but nitride layer depth is subject to the restriction of the small ball's diameter change simultaneously.While ensureing that the diameter of bead is in allowed band, improve bead surface hardness, setting nitride layer depth desired value is 0.05mm.See Fig. 1, as follows based on the concrete implementation step of the present invention:

Step one: determine test response (being paid close attention to mass property), factor of influence (comprising controllable factor and noise factor) and span thereof, the test total degree n that the restriction relation between each factor and test resource allow ^total.Here, consider test resource and schedule requirement, overall test number of times requires to be no more than 150 times.

By the degree of depth of nitration case responsively, in the process of ion soft-nitriding, all will experience a sputtering stage and two nitridation stage, table 1 summarizes each factor of influence and span thereof at every turn:

Table 1 factor of influence and span thereof

Theoretical according to ion soft-nitriding, for ensureing nitriding result, setting constraint condition: x ₅-x ₂>=20 DEG C, x ₉-x ₅>=10 DEG C.In addition, Ar, H ₂, N ₂throughput (L/min) is difficult to accurate control, and there is fluctuation, be noise factor, its tolerance is respectively ${\mathrm{\Δ}}_{x_3}=0.08L/min,{\mathrm{\Δ}}_{x_6}=0.1L/\min ,{\mathrm{\Δ}}_{x_7}=0.02L/min.$

Step 2: the horizontal number m adopted according to number and the plan of noise factor, selects suitable uniform designs table, and determines appearance test number (TN) n with this ^outer.System of selection brief introduction is as follows:

Suppose there be noise factor l, intend adopting number of levels m, then from uniform designs table storehouse, select evenly table according to deviation or and appearance test number (TN) is defined as n ^outer=ω.

Suppose there be noise factor l, intend adopting number of levels m, then select evenly to show U from uniform designs table storehouse _ω(m ^l), and appearance test number (TN) is defined as n ^outer=ω.

In this example, have 3 noise factors, for reducing test number (TN) under the prerequisite ensureing precision, each predictor selection 6 levels being tested, selects uniform designs table carry out appearance design, appearance test number (TN) n ^outer=6.

Step 3: determine to respond the relational model between the factor and interior watch test frequency n ^inner.

The concrete defining method of relational model in this step between response and the factor is as follows:

Based on the test essential information in step one, adopt the form of mathematical model to give expression to the relational model responded between the factor, be designated as y=X β+ε.Wherein, for known n ^inner× p ties up parameter model matrix; F (x _i) be about x _iknown function, react whole controllable factor and between restriction relation; β=(β ₀, β ₁, β ₂..., β _p) ^tfor p solve for parameter, react the interact relation between each factor and response.

Interior watch test frequency n ^innerdefining method as follows:

By general equivalence theorem, for the linear model having p parameter, exist by n ^*(p≤n ^*≤ p (p+1)/2) the D-optimal design ξ of individual testing site composition ^*.So, according to overall test frequency n ^totalwith appearance test number (TN) n ^outer, be not difficult to provide [p, the min (p (p+1)/2, n of watch test numbers range in D-optimum ^total/ n ^outer)].Therefore, according to actual test conditions, select any n ^inner∈ [p, min (p (p+1)/2, n ^total/ n ^outer)].

In the present embodiment, according to engineering experience, choose as drag:

y＝β ₀+x ₁β ₁+x ₂β ₂+xβ ₃+x ₄β ₄+x ₅β ₅+x ₆β ₆+x ₇β ₇+x ₈β ₈+x ₉β ₉+x ₂x ₃β ₂₃+

x ₂x ₅β ₂₅+x ₂x ₈β ₂₈+x ₃x ₅β ₃₅+x ₃x ₆β ₃₆+x ₃x ₇β ₃₇+x ₃x ₈β ₃₈+x ₃x ₉β ₃₉+x ₆x ₉β ₆₉+e

Wherein, error e ~ N (0, σ ²).

19 unknown parameters, i.e. p=19 are had in this model.In conjunction with interior table desired times, the span 19≤n of appearance test number (TN) can be determined ^inner≤ min (19 (19+1)/2, n ^total/ 6).Consider overall test resource, select interior watch test frequency n ^inner=19.

Step 4: in design section, adopts D-optimal-design method to provide given test number (TN) n ^innerinterior table design proposal, complete interior table design.Specific design method is as follows:

First, by each factor span specification in [-1,1] interval range, according to the restriction relation between the actual factor, design section is determined; Then, in this design section, select the design ξ meeting given number of times, make information matrix determinant is maximum, wherein, and f (x _i) be about x _iknown function, so far complete table design in D-optimum.Further, G-optimum efficiency is used to weigh the Optimality of this design.G-optimum efficiency is defined as:

$G_{eff}=\frac{p}{\underset{x\∈X}{max}d(x,\mathrm{\ξ})}$

Wherein, d (x, ξ)=f ^t(x) M ^-1(ξ) f (x).G-optimum efficiency value G _effmore close to 1, then represent that design ξ is better.

In the present embodiment, in D-optimum, table design table is as follows:

Table 2 is based on watch test scheme (coding) in D-optimal design

Sequence number	x 1	x 2	x 3	x 4	x 5	x 6	x 7	x 8	x 9
										1	-1	-1	-1	-1	-1	-1	1	-0.38418	1
2	-1	-0.12675	1	-1	0.66667	1	1	-1	1
										3	0.02211	-0.19135	0.001343	0.021786	-0.1083	-0.01131	0.050151	-0.03109	1
4	1	1	0.022368	1	0.66667	-0.00126	1	-0.01406	1
										5	-1	1	1	-1	-0.33333	-1	-1	1	1
6	1	0.31204	0.036341	-1	-1	1	1	-1	-0.66667
										7	-1	1	0.073784	1	0.66667	1	-1	1	1
8	1	-1	-1	-0.15154	-1	1	1	0.025892	1
										9	0.072806	-1	-1	1	-1	-0.03687	-1	-1	-0.66667
10	1	0.18165	1	1	-1	-1	1	-1	-0.66667
										11	-1	1	-1	-1	0.66667	-1	-0.10588	-0.045	1
12	-1	-1	0.020447	1	-0.09342	-1	-1	-1	0.23992
										13	-1	-0.03265	1	1	-1	1	1	1	1
14	1	-1	1	-1	-1	-1	1	-0.00697	0.27705
										15	1	-1	-0.26793	-1	-1	-1	-1	-1	1
16	-1	-1	1	1	0.66667	-1	-1	1	1

17	1	0.020626	0.074386	1	-1	-1	-1	0.055004	1
										18	-1	0.33333	1	-0.04891	-1	1	1	1	-0.66667
19	-0.08383	1	1	0.048094	0.66667	-1	-1	-1	1

The G efficiency of this design its design efficiency visible is higher.

Step 5: adopt based on quantile methods such as empirical distribution functions, for appearance testing site each in interior table, determines noise factor level, completes appearance design.

In this step, the defining method of noise factor level is as follows:

Note x _[i]for variable x based on quantile such as empirical distribution function i-th grade, it is defined as follows:

$x_{\[i\]}=\arg F\left(\frac{i}{m}\right),i=1,2,...,m$

Wherein, the distribution function that F (x) is stochastic variable x, m is noise factor number of levels; The inverse function of argF () representative function F ().In other words, above formula represents, when time, the value of corresponding stochastic variable x is designated as x _[i], x _[i]be i-th level value of factor x.

In this example, assuming that equal Normal Distribution N (0, the σ of noise factor ²), there is relationship delta=3 σ between tolerance Δ and standard deviation sigma, the variance that can obtain three noise factors is respectively: ${\mathrm{\σ}}_{x_3}^2={(0.08/3)}^2,{\mathrm{\σ}}_{x_6}^2={(0.1/3)}^2,{\mathrm{\σ}}_{x_7}^2={(0.02/3)}^2,$ Average μ then shows to determine in corresponding.Show in table 2 No. 2, three the noise factor number of levels determined under this condition are in table 3.

The setting of corresponding noise factor number of levels is shown in table No. 32

Then corresponding appearance is designed to:

Corresponding appearance design is shown in table No. 42

Step 6: specifically test according to testing program, according to testing the product performance index calculated mass characteristic signal to noise ratio (S/N ratio) obtained, matching obtains the regression function η (x between signal to noise ratio (S/N ratio) and the factor _i), calculate and make the maximized controllable factor optimum level of signal to noise ratio (S/N ratio).

In this example, the response nitride layer depth paid close attention to presents Definite purpose, according to gained test figure, calculates signal to noise ratio (S/N ratio) under each condition in table 5

Table 5 test design scheme (decoding) and signal to noise ratio (S/N ratio)

Utilize above-mentioned data, model parameter estimation is carried out to signal to noise ratio (S/N ratio) η and each factor, adopt method of gradual regression can obtain with drag

η＝3465.311+0.033862x ₁-1.34615x ₂+16.60486x ₃-0.1968x ₄-6.99205x ₅-384.59x ₆

+84.20613x ₇+0.3755x ₈-5.12495x ₉+0.148254x ₁x ₃+0.002458x ₂x ₅+0.328684x ₄x ₆

+0.255114x ₄x ₇+2.575394x ₆x ₈+0.88726x ₇x ₈+0.010374x ₅x ₉

Utilize non-linear solving method, obtain the factor level making signal to noise ratio (S/N ratio) maximum when satisfied test constraint and be combined as:

x ^*＝[30,500,0.6,120,520,0.07,0.475,150,530]

Step 7: based on Sensitivity Analysis Method determination stable factor and optimum level thereof.

Signal to noise ratio (S/N ratio) is to factor of influence x _i, i=1,2 ..., the sensitivity definition of n is as follows:

S _ijnumeric representation to variable x _isensitivity, S _ijsymbol represent to variable x _imonotonicity.The S that numerical value is larger _ijthe corresponding factor is stable factor, and get the combination of its optimal level with the stability reaching system, the fluctuation that system is exported is reduced to a minimum.Its complementary divisor is Dynamic gene, and the output valve of adjustment System makes it to reach or close to desired value, thus when reducing fluctuation as far as possible, obtains mass property is combined closest to the optimal parameter of its desired value.

In the present embodiment, the Calculation of Sensitivity of objective function η to each factor of influence is as follows:

The sensitivity of objective function η to all factors of influence arrogant to little gather for:

Table 6 η is to each factor of influence sensitivity summary sheet

Obviously, objective function η is to N ₂throughput is the most responsive, is secondly H ₂throughput and Ar throughput.Therefore, N is determined ₂, H ₂ar throughput three kinds of factors are stable factor, choose its optimum level to signal to noise ratio (S/N ratio); Determine that three phases duration of ventilation and ventilation air are Dynamic gene, utilize Dynamic gene that the mass property of design proposal is adjusted to desired value.

Step 8: regression fit tries to achieve unknown parameter in relational model y, and adjusts Dynamic gene according to this, determines optimum factor horizontal combination, to make under this testing program mass property estimated value closest to desired value.

In the present embodiment, utilize least square method, trying to achieve unknown parameters ' value in relational model y is:

β＝(β ₀,β ₁,...,β ₆₉)

＝(1.9666,0.0003,-0.086,-0.7368,-0.0002,-0.001,-0.0019,0.1384,0.78,

-0.0083,0,0572,-0.0006,0.0028,-0.0006,0.0011,-0.1771,0.001,0.003,0.231)

According to relational model, transfer Dynamic gene, determine that final optimal case is:

x ^*＝[30,440,0.6,120,350,0.07,0.475,120,500]

Under this parameter combinations, bead nitride layer depth predicted value is 5.01mm, reaches desired value requirement, satisfactorily completes the target of bead surface nitrogenize test design.

Claims (1)

1., based on the field mouth test design method showing in D-optimum to design, it is characterized in that: the method concrete steps are as follows:

Step one: according to test objective, condition and engineering experience, provide test fundamental test information, comprise that test response is paid close attention to mass property, factor of influence comprises controllable factor and noise factor and span thereof, the test total degree n that the restriction relation between each factor and test resource allow ^total;

Step 2: the horizontal number m adopted according to number and the plan of noise factor, selects suitable uniform designs table, and determines appearance test number (TN) n with this ^outer, system of selection brief introduction is as follows:

Suppose there be noise factor l, intend adopting number of levels m, then from uniform designs table storehouse, select evenly table appearance test number (TN) is defined as n simultaneously ^outer=ω;

Step 3: determine to respond the relational model between the factor and interior watch test frequency n ^inner;

The concrete defining method of relational model in this step between response and the factor is as follows:

Based on the test essential information in step one, adopt the form of mathematical model to give expression to the relational model responded between the factor, be designated as y=X β+ε; Wherein, for known n ^inner× p ties up parameter model matrix; F (x _i) be about x _iknown function, react whole controllable factor and between restriction relation; β=(β ₀, β ₁, β ₂..., β _p) ^tfor p solve for parameter, react the interact relation between each factor and response;

Interior watch test frequency n ^innerdefining method as follows:

By general equivalence theorem, for the linear model having p parameter, exist by n ^*(p≤n ^*≤ p (p+1)/2) the D-optimal design ξ of individual testing site composition ^*; So, according to overall test frequency n ^totalwith appearance test number (TN) n ^outer, be not difficult to provide [p, the min (p (p+1)/2, n of watch test numbers range in D-optimum ^total/ n ^outer)];

Therefore, according to actual test conditions, select any n ^inner∈ [p, min (p (p+1)/2, n ^total/ n ^outer)];

Step 4: in design section, adopts D-optimal-design method to provide given test number (TN) n ^innerinterior table design proposal, complete interior table design, its specific design method is as follows:

First, by each factor span specification in [-1,1] interval range, according to the restriction relation between the actual factor, design section is determined; Then, in this design section, select the design ξ meeting given number of times, make information matrix determinant is maximum, so far completes table design in D-optimum; Further, use G-optimum efficiency to weigh the Optimality of this design, G-optimum efficiency is defined as:

$G_{eff}=\frac{p}{\underset{x\∈X}{max}d(x,\mathrm{\ξ})}$

Here, G-optimum efficiency value G _effmore close to 1, represent that design ξ is better;

Step 5: adopt based on quantile methods such as empirical distribution functions, for appearance testing site each in interior table, determines noise factor level, completes appearance design;

In this step, the defining method of noise factor level is as follows:

Note x _[i]for variable x based on quantile such as empirical distribution function i-th grade, it is defined as follows:

$x_{\[i\]}=\arg F\left(\frac{i}{m}\right),i=1,2,...,m$

Wherein, the distribution function that F (x) is stochastic variable x, m is noise factor number of levels; The inverse function of argF () representative function F (); In other words, above formula represents, when time, the value of corresponding stochastic variable x is designated as x _[i], x _[i]be i-th level value of factor x;

Step 6: specifically test according to testing program, according to testing the product performance index calculated mass characteristic signal to noise ratio (S/N ratio) obtained, matching obtains the regression function η (x between signal to noise ratio (S/N ratio) and the factor _i), calculate and make the maximized controllable factor optimum level of signal to noise ratio (S/N ratio);

Step 7: based on Sensitivity Analysis Method determination stable factor and optimum level thereof;

Signal to noise ratio (S/N ratio) is to factor of influence x _i, i=1,2 ..., the sensitivity definition of n is as follows:

S _ijnumeric representation to variable x _isensitivity, S _ijsymbol represent to variable x _imonotonicity, the S that numerical value is larger _ijthe corresponding factor is stable factor, and get the combination of its optimal level with the stability reaching system, the fluctuation that system is exported is reduced to a minimum; Its complementary divisor is Dynamic gene, and the output valve of adjustment System makes it to reach or close to desired value, thus when reducing fluctuation as far as possible, obtains mass property is combined closest to the optimal parameter of its desired value;

Step 8: regression fit tries to achieve unknown parameter in relational model y, and adjusts Dynamic gene according to this, determines optimum factor horizontal combination, to make under this testing program mass property estimated value closest to desired value.