CN107330131A - The interval Optimization Method of component of machine parameters of structural dimension and its dimensional tolerance - Google Patents

The interval Optimization Method of component of machine parameters of structural dimension and its dimensional tolerance Download PDF

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CN107330131A
CN107330131A CN201610280903.9A CN201610280903A CN107330131A CN 107330131 A CN107330131 A CN 107330131A CN 201610280903 A CN201610280903 A CN 201610280903A CN 107330131 A CN107330131 A CN 107330131A
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CN107330131B (en
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李�荣
杨国来
葛建立
孙全兆
刘宁
李志旭
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Nanjing University of Science and Technology
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
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Abstract

The invention discloses the interval Optimization Method of a kind of component of machine parameters of structural dimension and its dimensional tolerance.Concretely comprise the following steps：It is determined that needing to carry out the parameters of structural dimension of the component of machine of interval optimum；Assuming that parameters of structural dimension Normal Distribution, discretization parameters of structural dimension and its function, set up the interval possibility degree model based on discretization；It is interval with interval midpoint and interval radius description scheme dimensional parameters, set up average tolerance grade factor；Set up the range optimization model of component of machine parameters of structural dimension and its dimensional tolerance；Penalty function method is converted into unconstrained optimization problem；Solved using multi-objective genetic algorithm.The present invention can optimize simultaneously obtains nominal size and dimensional tolerance, can optimize and obtain the more excellent solution of target capabilities, while average tolerance grade factor engineering significance is obvious.

Description

The interval Optimization Method of component of machine parameters of structural dimension and its dimensional tolerance
Technical field
The present invention relates to Optimal Design of Mechanical Structure field, specifically a kind of interval possibility degree and processing based on discretization The component of machine parameters of structural dimension and the interval Optimization Method of dimensional tolerance of precision.
Background technology
Traditional mechanical design challenges are generally based on the parameter and Optimized model of determination, and by means of classical certainty Optimization problem is solved, but physical dimension, material property, load, boundary condition, component in actual engineering problem The error or uncertainty of parameter, measured deviation etc. are certainly existed, these errors or the uncertain optimization mesh that can influence to design Mark performance or change the feasibility of constraint, the optimization that the present invention starts with to carry out mechanical structure from the uncertainty of physical dimension is set Meter.
Traditional structural dimension optimization design is the optimal solution obtained based on deterministic models, can optimize and obtain design parameter Nominal size, further according to practical engineering experience give nominal size tolerance, the size of actual parameter can be attached in nominal size Near to float, scale error is that the uncertain factor brought under prior art conditions by manufacture, measurement, assembling and abrasion etc. is made Into it can cause target capabilities and constraints to fluctuate within the specific limits, and particularly critical feature size parameter error can shadow Ring the optimization aim performance of design or change the feasibility of constraint, now must take into consideration target capabilities interval and constraint condition portion Between.The distribution of critical feature size parameter error can be obtained by long-term production practices, but interval point of target capabilities Unknown Distribution between cloth and constraint condition portion, but will be to the mesh of different critical feature size parameter error schemes in optimization process Mark performance interval is compared simultaneously so that constraints interval is met.
Recent domestic scholar proposes many methods for parameter uncertainty, it is however generally that these methods are based on three Class model：Probabilistic model, fuzzy model, non-probability model.Probabilistic model, will be uncertain based on probability theory and stochastic programming Property parameter regard stochastic variable as, pass through statistical method construct uncertain parameters exact probability be distributed.Fuzzy model is with fuzzy Based on theoretical and fuzzy programming, regard uncertain parameter as fuzzy number, construct its fuzzy membership function.Non-probability model without Accurately probability distribution or fuzzy membership function need to be constructed, but uncertain parameters are described with convex set or interval, The bound of uncertain parameters only need to be known.
C.Jiang, H.C.Xie, Z.G.Zhang propose a kind of bounded-but-unknown uncertainty optimization method for considering tolerance, fixed One nondimensional design tolerance index of justice, constraint function is handled using equally distributed interval possibility degree model is obeyed, this The method of kind has obtained nominal size and tolerance, but it only tries to achieve average criterion performance with design variable interval midpoint, not by mesh Mark performance considers into interval, and the interval distribution of target capabilities is not accounted for more, while the dimensionless design tolerance index office of definition It is limited to mathematical meaning.
The content of the invention
The present invention is unable to Synchronous fluorimetry, critical feature size parameter error meeting to solve parameters of structural dimension and dimensional tolerance There is provided a kind of component of machine physical dimension for the feasibility problems that the optimization aim performance or change for influenceing structure design are constrained The interval Optimization Method of parameter and its dimensional tolerance, is discrete random variable by the interval discretization of critical feature size parameter, To obtain the distribution of target capabilities interval, average tolerance grade factor evaluation function is built by evaluation index of machining accuracy, is set up Consider that the uncertainty optimization model of the distribution of target capabilities interval, the distribution of constraints interval and machining accuracy optimizes key simultaneously The nominal size and dimensional tolerance of parameters of structural dimension.
The technology of the present invention solution：A kind of range optimization side of component of machine parameters of structural dimension and its dimensional tolerance Method, comprises the following steps：
Step 1: determining to need to carry out the parameters of structural dimension set X=[x of the component of machine of interval optimum1, x2,...,xm], m is the number of parameter, the parameters of structural dimension x containing dimensional tolerancejFor：
Step 2: assuming parameters of structural dimensionNormal Distribution, sets up parameters of structural dimension xjLess than or equal to size a Interval possibility degree；By parameters of structural dimension discretization, the value and value for obtaining discrete structure dimensional parameters are corresponding general Rate；Discrete structure dimensional parameters are substituted into function f (X) again, function f (X) discretization is realized；Set up the area based on discretization Between possibility degree model；Set up and compared two-by-two on parameters of structural dimension interval scheme by the interval possibility degree model based on discretization Compared with Possibility Degree Matrix；
Step 3: it is interval with interval midpoint and interval radius description scheme dimensional parameters, set up average tolerance grade factor A；
Step 4: uncertain excellent with the interval possibility degree model based on discretization and average tolerance grade factor A conversions Change problem is deterministic optimization problem, sets up the range optimization model of component of machine parameters of structural dimension and its dimensional tolerance；
Step 5: with penalty function method by the constrained range optimization model conversation of step 4 be unconfined range optimization Model；
Step 6: with the unconfined range optimization model of multi-objective genetic algorithm solution procedure five.
Compared with prior art, its remarkable advantage is the present invention：
(1) compared with traditional structural optimization problems, the interval Optimization Method set up not only obtains physical dimension Nominal size, and its dimensional tolerance is also obtained, greatly shortens the design cycle cost-effective.
(2) compared with traditional range optimization, the optimization method set up considers the distribution of object function interval, constraint bar The distribution of part interval, target capabilities are more excellent in practical situations both for obtained optimizing design scheme, and the feasibility of constraint is more reliable.
(3) consider that the average tolerance grade factor engineering significance that machining accuracy is set up is obvious so that Optimized model is considering On the basis of target capabilities and constraints, it is contemplated that actual processing precision factor so that the result of optimization more meets actual feelings Condition.
Brief description of the drawings
The flow chart of Fig. 1 interval Optimization Methods of the present invention.
Fig. 2 combines the genetic algorithm flow charts of NGSA- II of interval Optimization Method of the present invention.
The optimal interval expectation of the object function of Fig. 3 embodiments 1 with iterations curve.
The optimum value of the average grade of tolerance evaluation function of Fig. 4 embodiments 1 with iterations curve.
The target capabilities and average tolerance grade factor of the optimal Pareto disaggregation of Fig. 5 embodiments 1.
The optimal interval expectation of the object function of Fig. 6 embodiments 2 with iterations curve.
The optimum value of the average grade of tolerance evaluation function of Fig. 7 embodiments 2 with iterations curve.
The target capabilities and average tolerance grade factor of the optimal Pareto disaggregation of Fig. 8 embodiments 2.
Embodiment
The present invention constructs a kind of component of machine structure based on the interval possibility degree of discretization and average tolerance grade factor The interval Optimization Method of dimensional parameters and its dimensional tolerance.
The present invention will be further described with reference to the accompanying drawings and examples.
A kind of component of machine parameters of structural dimension and its dimensional tolerance with reference to described in Fig. 1 present embodiments it is interval excellent The foundation of change method is comprised the following steps that：
Step 1: determining to need to carry out m X=[x of parameters of structural dimension of the component of machine of interval optimum1, x2,...,xm], parameters of structural dimension xjIt is expressed as after considering dimensional tolerance：
Step 2: assuming parameters of structural dimensionNormal Distribution, sets up parameters of structural dimension xjLess than or equal to size a Interval possibility degree；By parameters of structural dimension discretization, the value and value for obtaining discrete structure dimensional parameters are corresponding general Rate；Discrete structure dimensional parameters are substituted into function f (X) again, function f (X) discretization is realized；Set up the area based on discretization Between possibility degree model；Set up and compared two-by-two on parameters of structural dimension interval scheme by the interval possibility degree model based on discretization Compared with Possibility Degree Matrix；
Step 2 (one), assume parameters of structural dimension obey withFor average,For standard deviation Normal distribution, the interval of Normal DistributionInterval possibility degree model less than or equal to size a is expressed as follows：
Φ (x) is the distribution function of the standardized normal distribution of μ=0, σ=1 in formula；
It is step 2 (two), parameters of structural dimension is intervalIt is divided into n equal portions, every part is a unit, Mei Gedan Member has two nodesSimultaneously comprising a unit midpointWith unit midpoint come instead of each unit, unit correspondence Probability be exactly unit midpoint probability, be expressed as follows：
The corresponding possibility degree of each level value, by formula (4), possibility degree model is obtained：
Parameters of structural dimensionIt is discrete be discrete random variable after be denoted as<xj>、p(<xj>)；
Assuming that separate between parameters of structural dimension, the horizontal number of value during each parameters of structural dimension discretization is equal For n, a m dimension design parameter matrix is constituted<X>n×n×...×n, each element is designated as
M ties up design parameter matrix<X>n×n×...×nEach the corresponding possibility degree of element constitutes m dimension Possibility Degree Matrixes<P >n×n×...×n, the corresponding possibility degree of each element had using the Joint Distribution formula of separate discrete variable：
WillFunction f (X) is substituted into obtainFormation m dimension Jacobian matrixs f (<X >)n×n×...×n, its corresponding m dimension Possibility Degree Matrix is still<P>n×n×...×n；
Step 2 (three), interval possibility degree model of the foundation based on discretization：
f(<Xj>)n×n×...×nElement be less than or equal to f (<Xz>)n×n×...×nElement possibility degree be：
By f (<Xj>)n×n×...×nEach element and f (<Xz>)n×n×...×nAll elements by formula (6) calculate, then Summation, f (Xj)ILess than or equal to f (Xz)IInterval possibility degree model be：
Step 2 (four), set up the Possibility Degree Matrix that parameters of structural dimension interval scheme compares two-by-two；
Parameters of structural dimension interval schemeCorresponding object function forms m dimension Jacobian matrixs respectively f(<X1>)n×n×...×n, f (<X2>)n×n×...×n..., f (<Xq>)n×n×...×n；Parameters of structural dimension interval scheme compares two-by-two Possibility Degree Matrix is：
Wherein pjk=p (f (Xj)I≤f(Xk)I)；J, k=1,2 ..., q.
Each row element of formula (8) is added up and obtained：
Pi=pi1+pi2+...+pi(j-1)+pi(j+1)+...+piq；I, j=1,2 ..., q. (9)
When optimization problem finds a function f (X) minimum value, with-PjIt is used as sort by.
Step 3: it is interval with interval midpoint and interval radius description scheme dimensional parameters, set up average tolerance grade factor A；
It is step 3 (one), interval with interval midpoint and interval radius description scheme dimensional parameters：
In formulaFor interval midpoint,For interval radius；
Step 3 (two), the average tolerance grade factor evaluation function of foundation：
First set up the tolerance grade factor of continuous type：
In formulaWherein D is diameter, unit mm；I is tolerance Unit, unit mm；
Average tolerance grade factor evaluation function is：
Step 4: uncertain excellent with the interval possibility degree model based on discretization and average tolerance grade factor A conversions Change problem is deterministic optimization problem, sets up the range optimization model of component of machine parameters of structural dimension and its dimensional tolerance；
General structural optimization problems are converted into following Internal optimum problem, because optimization problem is minimized, average Add negative sign before tolerance grade factor, then the range optimization model of component of machine parameters of structural dimension and its dimensional tolerance is：
L is the number of constraint function in formula.
Step 5: with penalty function method by the constrained range optimization model conversation of step 4 be unconfined range optimization mould Type：
By Part I object function min f (<XC, XW>)n×n×...×nWith constraint function gj(<XC, XW>)n×n×...×nCorrespondence The element of position is handled using penalty function obtains augmented objective function 1：
M is penalty factor in formula；
By all elements after constraint function discretization, with penalty function method, all the average grade of tolerance of Part II is arrived in punishment On factor evaluation function A, augmented objective function 2 is obtained：
Constrained Internal optimum problem (13) is converted into by formula (14), (15) following without constraint Internal optimum problem：
Step 6: with the unconfined component of machine parameters of structural dimension of multi-objective genetic algorithm solution procedure five and its The range optimization model of dimensional tolerance.
Embodiment 1：
The following certainty optimization problem containing two design parameters：
According to formula (5), n takes 10, obtained m dimension Possibility Degree Matrixes<P>10×10For：
According to formula (13), above-mentioned optimization problem is converted into：
Optimal design parameter in above formula is
Unconstrained optimization problem is converted into formula (16), with reference to Fig. 2, is calculated using the Progran of Genetlc Algorithm of NSGA- II, population Number is 100, the generation of iteration 400, and the crossover probability for the intersection that counts is 0.8, and the mutation probability of Gaussian mutation is 0.1.
Knowing with reference to Fig. 3, Fig. 4 before the generation of genetic algorithm iteration 220 has repeatedly substantially jump, illustrates to have searched out more excellent individual, Target capabilities and average tolerance grade factor do not change substantially after 220 generations, illustrate algorithmic statement.
With reference to Fig. 4, obtained optimal Pareto disaggregation, the optimum interval for reflecting target capabilities can be with average tolerance etc. The increase of level coefficient and be gradually deteriorated, both are into negative correlation.
With reference to table 1, key design parameter is no longer the value of a determination, but comprising error interval；Obtain The interval of object function and distribution, the average tolerance grade factor of each design.
The part Pareto optimal solutions of the embodiment 1 of table 1.
Embodiment 2
Big, the problem of structure is not compact enough, herein using above-mentioned excellent for the recoil absorber radial dimension of certain large caliber gun Change model to obtain its parameter error scheme.Recoil absorber main structure parameters have recoil absorber active length L, piston area A0, recoil cylinder internal diameter DT, system move back bar outside diameter dT, system move back bar intracavity diameter d1, restraining ring diameter dp, recoil cylinder outer diameter D1, wherein its His parameter all relies on recoil cylinder internal diameter DTBar outside diameter d is moved back with systemTObtain, choose DTAnd dTFor key design parameter [x1, x2]= [DT, dT]；The piston that object function obtains for the piston area that consideration practical structures are obtained with consideration recoil absorber liquid measure temperature rise The absolute value of the difference of work area is minimum, can obtain the piston area of minimum so that recoil absorber radial dimension is as far as possible small, knot Structure is compacter；Constraints 1 makes the strength condition moved back when bar enters again by chamber pressure to meet, by the thick cyclinder by internal pressure Formula；Constraints 2 is to meet graduating stem stability condition, the stable Euler's formula of pressing lever；Boundary condition is key Design The upper lower limit value of parameter.
The optimized mathematical model is written as：
Wherein E is free recoil energy, and α is estimation coefficient, λmaxFor maximum recoil length, e is consideration rigging error and penetrates The surplus hit condition and retained；hmTo enter governor groove depth capacity, σ againsThe yield limit of bar material is moved back for system；N is safety Coefficient, λjxLong, the F for limit recoilΦfmaxTo enter governor maximum hydraulic pressure resistance again, K is for relevant with bar two ends fixing situation Number, graduating stem, which is considered as one end, to be fixed one end and is hinged K=2 π, EsFor graduating stem elasticity modulus of materials.
According to formula (13), while n takes 10, above-mentioned optimization problem is converted into：
Optimization design variable in above formula is
Unconstrained optimization problem is converted into formula (16), is calculated with reference to Fig. 2, and using the Progran of Genetlc Algorithm of NSGA- II.
Knowing with reference to Fig. 6, Fig. 7 before the generation of genetic algorithm iteration 150 has repeatedly substantially jump, illustrates to have searched out more excellent individual, Target capabilities and average tolerance grade factor do not change substantially after 150 generations, illustrate algorithmic statement.
With reference to Fig. 4, obtained optimal Pareto disaggregation, the optimum interval for reflecting target capabilities can be with average tolerance etc. The increase of level coefficient and be gradually deteriorated, both are into negative correlation.
With reference to table 2, it is no longer a value determined that recoil cylinder internal diameter and system, which move back bar external diameter, but comprising error area Between；Interval and the distribution of object function, the average tolerance grade factor of each design are obtained.
The part Pareto optimal solutions of the embodiment 2 of table 2

Claims (4)

1. the interval Optimization Method of a kind of component of machine parameters of structural dimension and its dimensional tolerance, it is characterised in that including following Step：
Step 1: determining to need to carry out the parameters of structural dimension set X=[x of the component of machine of interval optimum1, x2,...,xm], m is the number of parameter, the parameters of structural dimension x containing dimensional tolerancejFor：
<mrow> <msubsup> <mi>x</mi> <mi>j</mi> <mi>I</mi> </msubsup> <mo>=</mo> <mo>&amp;lsqb;</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>L</mi> </msubsup> <mo>,</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>R</mi> </msubsup> <mo>&amp;rsqb;</mo> <mo>=</mo> <mo>{</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>|</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>L</mi> </msubsup> <mo>&amp;le;</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>&amp;le;</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>R</mi> </msubsup> <mo>,</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>&amp;Element;</mo> <mi>R</mi> <mo>}</mo> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>m</mi> <mo>.</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
Step 2: assuming parameters of structural dimensionNormal Distribution, sets up parameters of structural dimension xjLess than or equal to size a area Between possibility degree；By parameters of structural dimension discretization, the corresponding probability of value and value of discrete structure dimensional parameters is obtained；Again Discrete structure dimensional parameters are substituted into function f (X), function f (X) discretization is realized；Setting up the interval based on discretization can Energy degree model；Set up what is compared two-by-two on parameters of structural dimension interval scheme by the interval possibility degree model based on discretization Possibility Degree Matrix；
Step 3: it is interval with interval midpoint and interval radius description scheme dimensional parameters, set up average tolerance grade factor A；
Step 4: being asked with the interval possibility degree model based on discretization and average tolerance grade factor A conversions uncertainty optimization Entitled deterministic optimization problem, sets up the range optimization model of component of machine parameters of structural dimension and its dimensional tolerance；
Step 5: with penalty function method by the constrained range optimization model conversation of step 4 be unconfined range optimization mould Type；
Step 6: with the unconfined range optimization model of multi-objective genetic algorithm solution procedure five.
2. the interval Optimization Method of component of machine parameters of structural dimension according to claim 1 and its dimensional tolerance, its It is characterised by：The step 2 concrete methods of realizing is as follows：
Step 2 (one), assume parameters of structural dimension obey withFor average,For the normal state of standard deviation Distribution, the interval of Normal DistributionInterval possibility degree model less than or equal to size a is expressed as follows：
<mrow> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>&amp;le;</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mn>0</mn> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>a</mi> <mo>&lt;</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>L</mi> </msubsup> <mo>;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>-</mo> <msub> <mi>&amp;mu;</mi> <mi>j</mi> </msub> </mrow> <msub> <mi>&amp;sigma;</mi> <mi>j</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <mo>-</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <mo>-</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>x</mi> <mi>j</mi> <mi>L</mi> </msubsup> <mo>&amp;le;</mo> <mi>a</mi> <mo>&amp;le;</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>R</mi> </msubsup> <mo>;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>1</mn> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>a</mi> <mo>&gt;</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>R</mi> </msubsup> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
Φ (x) is the distribution function of the standardized normal distribution of μ=0, σ=1 in formula；
It is step 2 (two), parameters of structural dimension is intervalIt is divided into n equal portions, every part is a unit, and each unit has two Individual nodeSimultaneously comprising a unit midpointWith unit midpoint come instead of each unit, the corresponding probability of unit For the probability at unit midpoint, it is expressed as follows：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mrow> <mi>j</mi> <mi>k</mi> </mrow> <mi>L</mi> </msubsup> <mo>=</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>L</mi> </msubsup> <mo>+</mo> <mfrac> <mrow> <msubsup> <mi>x</mi> <mi>j</mi> <mi>R</mi> </msubsup> <mo>-</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>L</mi> </msubsup> </mrow> <mi>n</mi> </mfrac> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mrow> <mi>j</mi> <mi>k</mi> </mrow> <mi>R</mi> </msubsup> <mo>=</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>L</mi> </msubsup> <mo>+</mo> <mfrac> <mrow> <msubsup> <mi>x</mi> <mi>j</mi> <mi>R</mi> </msubsup> <mo>-</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>L</mi> </msubsup> </mrow> <mi>n</mi> </mfrac> <mi>k</mi> <mo>;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mrow> <mi>j</mi> <mi>k</mi> </mrow> <mi>c</mi> </msubsup> <mo>=</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>L</mi> </msubsup> <mo>+</mo> <mfrac> <mrow> <msubsup> <mi>x</mi> <mi>j</mi> <mi>R</mi> </msubsup> <mo>-</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>L</mi> </msubsup> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mtable> <mtr> <mtd> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>m</mi> <mo>.</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
The corresponding possibility degree of each level value, by formula (4), possibility degree model is obtained：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>=</mo> <msubsup> <mi>x</mi> <mrow> <mi>j</mi> <mi>k</mi> </mrow> <mi>c</mi> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>&amp;le;</mo> <msubsup> <mi>x</mi> <mrow> <mi>j</mi> <mi>k</mi> </mrow> <mi>R</mi> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <mi>P</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>&amp;le;</mo> <msubsup> <mi>x</mi> <mrow> <mi>j</mi> <mi>k</mi> </mrow> <mi>L</mi> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <mo>-</mo> <mn>3</mn> <mo>+</mo> <mfrac> <mn>6</mn> <mi>n</mi> </mfrac> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <mo>-</mo> <mn>3</mn> <mo>+</mo> <mfrac> <mn>6</mn> <mi>n</mi> </mfrac> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <mo>-</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>m</mi> <mo>;</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
Parameters of structural dimensionIt is discrete be discrete random variable after be denoted as<xj>、p(<xj>)；
Assuming that separate between parameters of structural dimension, the horizontal number of value during each parameters of structural dimension discretization is n, Constitute a m dimension design parameter matrix<X>n×n×...×n, each element is designated as
M ties up design parameter matrix<X>n×n×...×nEach the corresponding possibility degree of element constitutes m dimension Possibility Degree Matrixes<P>n×n×...×n, The corresponding possibility degree of each element is had using the Joint Distribution formula of separate discrete variable：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>...</mo> <msub> <mi>k</mi> <mi>m</mi> </msub> </mrow> </msub> <mo>=</mo> <mi>p</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>=</mo> <msubsup> <mi>x</mi> <mrow> <mn>1</mn> <msub> <mi>k</mi> <mn>1</mn> </msub> </mrow> <mi>c</mi> </msubsup> <mo>)</mo> </mrow> <mo>&amp;times;</mo> <mi>p</mi> <mo>,</mo> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <msubsup> <mi>x</mi> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mn>2</mn> </msub> </mrow> <mi>c</mi> </msubsup> <mo>)</mo> </mrow> <mo>&amp;times;</mo> <mo>...</mo> <mo>&amp;times;</mo> <mi>p</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>m</mi> </msub> <mo>=</mo> <msubsup> <mi>x</mi> <mrow> <msub> <mi>mk</mi> <mi>m</mi> </msub> </mrow> <mi>c</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>k</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
WillFunction f (X) is substituted into obtainFormation m dimension Jacobian matrixs f (<X>)n×n×...×n, its is right The m answered ties up Possibility Degree Matrix<P>n×n×...×n；
Step 2 (three), interval possibility degree model of the foundation based on discretization：
f(<Xj>)n×n×...×nElement be less than or equal to f (<Xz>)n×n×...×nElement possibility degree be：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>k</mi> <mn>2</mn> </msub> <mn>...</mn> <msub> <mi>k</mi> <mi>m</mi> </msub> <msub> <mi>h</mi> <mn>1</mn> </msub> <msub> <mi>h</mi> <mn>2</mn> </msub> <mn>...</mn> <msub> <mi>h</mi> <mi>m</mi> </msub> </mrow> </msub> <mo>=</mo> <mi>p</mi> <mrow> <mo>(</mo> <mi>f</mi> <mo>(</mo> <mrow> <mo>&lt;</mo> <msub> <mi>X</mi> <mi>j</mi> </msub> <msub> <mo>&gt;</mo> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>k</mi> <mn>2</mn> </msub> <mn>...</mn> <msub> <mi>k</mi> <mi>m</mi> </msub> </mrow> </msub> </mrow> <mo>)</mo> <mo>&amp;le;</mo> <mi>f</mi> <mo>(</mo> <mrow> <mo>&lt;</mo> <msub> <mi>X</mi> <mi>z</mi> </msub> <msub> <mo>&gt;</mo> <mrow> <msub> <mi>h</mi> <mn>1</mn> </msub> <msub> <mi>h</mi> <mn>2</mn> </msub> <mn>...</mn> <msub> <mi>h</mi> <mi>m</mi> </msub> </mrow> </msub> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>=</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>k</mi> <mn>2</mn> </msub> <mn>...</mn> <msub> <mi>k</mi> <mi>m</mi> </msub> </mrow> </msub> <mo>&amp;times;</mo> <msub> <mi>p</mi> <mrow> <msub> <mi>h</mi> <mn>1</mn> </msub> <msub> <mi>h</mi> <mn>2</mn> </msub> <mn>...</mn> <msub> <mi>h</mi> <mi>m</mi> </msub> </mrow> </msub> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mo>&lt;</mo> <msub> <mi>X</mi> <mi>j</mi> </msub> <msub> <mo>&gt;</mo> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>k</mi> <mn>2</mn> </msub> <mn>...</mn> <msub> <mi>k</mi> <mi>m</mi> </msub> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>&amp;le;</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mo>&lt;</mo> <msub> <mi>X</mi> <mi>z</mi> </msub> <msub> <mo>&gt;</mo> <mrow> <msub> <mi>h</mi> <mn>1</mn> </msub> <msub> <mi>h</mi> <mn>2</mn> </msub> <mn>...</mn> <msub> <mi>h</mi> <mi>m</mi> </msub> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>0</mn> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mo>&lt;</mo> <msub> <mi>X</mi> <mi>j</mi> </msub> <msub> <mo>&gt;</mo> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>k</mi> <mn>2</mn> </msub> <mn>...</mn> <msub> <mi>k</mi> <mi>m</mi> </msub> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>&gt;</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mo>&lt;</mo> <msub> <mi>X</mi> <mi>z</mi> </msub> <msub> <mo>&gt;</mo> <mrow> <msub> <mi>h</mi> <mn>1</mn> </msub> <msub> <mi>h</mi> <mn>2</mn> </msub> <mn>...</mn> <msub> <mi>h</mi> <mi>m</mi> </msub> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>j</mi> <mo>,</mo> <mi>z</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>m</mi> <mo>;</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>k</mi> <mi>m</mi> </msub> <mo>,</mo> <msub> <mi>h</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>h</mi> <mn>2</mn> </msub> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>h</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>n</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
By f (<Xj>)n×n×...×nEach element and f (<Xz>)n×n×...×nAll elements by formula (6) calculate, Ran Houqiu With f (Xj)ILess than or equal to f (Xz)IInterval possibility degree model be：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>p</mi> <mrow> <mo>(</mo> <mi>f</mi> <msup> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mi>I</mi> </msup> <mo>&lt;</mo> <mi>f</mi> <msup> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>z</mi> </msub> <mo>)</mo> </mrow> <mi>I</mi> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>p</mi> <mrow> <mo>(</mo> <mi>f</mi> <msub> <mrow> <mo>(</mo> <mrow> <mo>&lt;</mo> <msub> <mi>X</mi> <mi>j</mi> </msub> <mo>&gt;</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>&amp;times;</mo> <mi>n</mi> <mo>&amp;times;</mo> <mn>...</mn> <mo>&amp;times;</mo> <mi>n</mi> </mrow> </msub> <mo>&lt;</mo> <mi>f</mi> <msub> <mrow> <mo>(</mo> <mrow> <mo>&lt;</mo> <msub> <mi>X</mi> <mi>z</mi> </msub> <mo>&gt;</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>&amp;times;</mo> <mi>n</mi> <mo>&amp;times;</mo> <mn>...</mn> <mo>&amp;times;</mo> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mn>...</mn> <munderover> <mo>&amp;Sigma;</mo> <mrow> <msub> <mi>k</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <msub> <mi>h</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <msub> <mi>h</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mn>...</mn> <munderover> <mo>&amp;Sigma;</mo> <mrow> <msub> <mi>h</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>p</mi> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>k</mi> <mn>2</mn> </msub> <mn>...</mn> <msub> <mi>k</mi> <mi>m</mi> </msub> <msub> <mi>h</mi> <mn>1</mn> </msub> <msub> <mi>h</mi> <mn>2</mn> </msub> <mn>...</mn> <msub> <mi>h</mi> <mi>m</mi> </msub> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
Step 2 (four), set up the Possibility Degree Matrix that parameters of structural dimension interval scheme compares two-by-two；
Parameters of structural dimension interval schemeCorresponding object function formed respectively m dimensions Jacobian matrix f (<X1 >)n×n×...×n, f (<X2>)n×n×...×n..., f (<Xq>)n×n×...×n；The possibility that parameters of structural dimension interval scheme compares two-by-two Spending matrix is：
Wherein pjk=p (f (Xj)I≤f(Xk)I)；J, k=1,2 ..., q；
Each row element of formula (8) is added up and obtained：
Pi=pi1+pi2+...+pi(j-1)+pi(j+1)+...+piq；I, j=1,2 ..., q. (9)
When optimization problem finds a function f (X) minimum value, with-PjIt is used as sort by.
3. the interval Optimization Method of component of machine parameters of structural dimension according to claim 1 and its dimensional tolerance, its It is characterised by：The step 3 concrete methods of realizing is as follows：
It is step 3 (one), interval with interval midpoint and interval radius description scheme dimensional parameters：
<mrow> <msubsup> <mi>x</mi> <mi>j</mi> <mi>I</mi> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>c</mi> </msubsup> <mo>,</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>w</mi> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mo>{</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>|</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>c</mi> </msubsup> <mo>-</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>w</mi> </msubsup> <mo>&amp;le;</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>&amp;le;</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>c</mi> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>w</mi> </msubsup> <mo>,</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>&amp;Element;</mo> <mi>R</mi> <mo>}</mo> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>m</mi> <mo>.</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
In formulaFor interval midpoint,For interval radius；
Step 3 (two), the average tolerance grade factor evaluation function of foundation：
First set up the tolerance grade factor of continuous type：
<mrow> <msub> <mi>a</mi> <mi>j</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msubsup> <mi>x</mi> <mi>j</mi> <mi>w</mi> </msubsup> </mrow> <mi>i</mi> </mfrac> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>2</mn> <mo>&amp;times;</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> <mo>&amp;times;</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>w</mi> </msubsup> </mrow> <mrow> <mn>4.5</mn> <mo>&amp;times;</mo> <mroot> <msubsup> <mi>x</mi> <mi>j</mi> <mi>c</mi> </msubsup> <mn>3</mn> </mroot> <mo>&amp;times;</mo> <mn>0.01</mn> <msubsup> <mi>x</mi> <mi>j</mi> <mi>c</mi> </msubsup> </mrow> </mfrac> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>x</mi> <mi>j</mi> <mi>c</mi> </msubsup> <mo>&amp;le;</mo> <mn>500</mn> <mi>m</mi> <mi>m</mi> <mo>;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>2</mn> <mo>&amp;times;</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> <mo>&amp;times;</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>w</mi> </msubsup> </mrow> <mrow> <mn>0.04</mn> <msubsup> <mi>x</mi> <mi>j</mi> <mi>c</mi> </msubsup> <mo>+</mo> <mn>21</mn> </mrow> </mfrac> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mn>500</mn> <mo>&lt;</mo> <msubsup> <mi>x</mi> <mi>j</mi> <mi>c</mi> </msubsup> <mo>&amp;le;</mo> <mn>3150</mn> <mi>m</mi> <mi>m</mi> <mo>;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>m</mi> <mo>.</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
In formulaWherein D is diameter, unit mm；I is allowance unit, Unit mm；
Average tolerance grade factor evaluation function is：
<mrow> <mi>A</mi> <mo>=</mo> <mroot> <mrow> <munderover> <mo>&amp;Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msub> <mi>a</mi> <mi>j</mi> </msub> </mrow> <mi>m</mi> </mroot> <mo>.</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
4. the interval Optimization Method of component of machine parameters of structural dimension according to claim 1 and its dimensional tolerance, its It is characterised by：The step 4 concrete methods of realizing is as follows：
General structural optimization problems are converted by the interval possibility degree model based on discretization and average tolerance grade factor For following Internal optimum problem, because optimization problem is minimized, negative sign is added before average tolerance grade factor, then machinery zero The range optimization model of part physical dimension and its dimensional tolerance is：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mrow> <mo>(</mo> <msup> <mi>X</mi> <mi>c</mi> </msup> <mo>,</mo> <msup> <mi>X</mi> <mi>w</mi> </msup> <mo>)</mo> </mrow> </munder> <mo>&amp;lsqb;</mo> <mi>f</mi> <msub> <mrow> <mo>(</mo> <mo>&lt;</mo> <msup> <mi>X</mi> <mi>C</mi> </msup> <mo>,</mo> <msup> <mi>X</mi> <mi>W</mi> </msup> <mo>&gt;</mo> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>&amp;times;</mo> <mi>n</mi> <mo>&amp;times;</mo> <mo>...</mo> <mo>&amp;times;</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>-</mo> <mi>A</mi> <mrow> <mo>(</mo> <msup> <mi>X</mi> <mi>C</mi> </msup> <mo>,</mo> <msup> <mi>X</mi> <mi>W</mi> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <mrow> <msub> <mi>g</mi> <mi>&amp;alpha;</mi> </msub> <msub> <mrow> <mo>(</mo> <mo>&lt;</mo> <msup> <mi>X</mi> <mi>C</mi> </msup> <mo>,</mo> <msup> <mi>X</mi> <mi>W</mi> </msup> <mo>&gt;</mo> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>&amp;times;</mo> <mi>n</mi> <mo>&amp;times;</mo> <mo>...</mo> <mo>&amp;times;</mo> <mi>n</mi> </mrow> </msub> <mo>&lt;</mo> <mn>0</mn> <mo>,</mo> <mi>&amp;alpha;</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>l</mi> <mo>;</mo> </mrow> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>X</mi> <mi>L</mi> </msub> <mo>&lt;</mo> <mrow> <mo>(</mo> <msup> <mi>X</mi> <mi>C</mi> </msup> <mo>,</mo> <msup> <mi>X</mi> <mi>W</mi> </msup> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi>X</mi> <mi>R</mi> </msub> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
L is the number of constraint function in formula.
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