KR102105032B1 - An efficient pareto set selection method for optimization of multi-objective systems - Google Patents

Abstract

The present invention is a method for selecting a set of pareto based on at least two or more performance indicators by a computer processor, comprising: a first simulation resource allocation step; A second simulation resource allocation step of calculating the uncertainty value of the design using statistical information and updating the statistical information by additionally allocating an extra simulation resource corresponding to the uncertainty value; And presenting a design plan included in the Pareto set through the updated statistical information. It may include.

Description

Efficient Pareto Set Selection Method for Optimizing Multiple Performances of a System {AN EFFICIENT PARETO SET SELECTION METHOD FOR OPTIMIZATION OF MULTI-OBJECTIVE SYSTEMS}

The present invention relates to a method for selecting a pareto set using a probabilistic simulation technique, and more specifically, to a method for efficiently selecting a pareto set through a probabilistic simulation technique for optimization of a system having at least two performance indicators. .

According to the 4th industrial revolution, modeling and simulation technology has emerged as a future technology and has been applied to various fields. Especially in the field of system design, probabilistic simulation is useful for accurately evaluating and optimizing the performance of complex systems that cannot be represented by mathematically closed-form equations such as supply chain, transportation, and manufacturing. However, in order to obtain accurate performance, it is required to perform iterative simulation to obtain a statistically significant value (steady-state value) for each system design.

In recent years, as the system becomes more complex, the cost of performing simulation, such as computing time, has also increased rapidly. In addition, as the number of design targets for performance evaluation increases, the simulation cost may increase. As a result, in order to efficiently apply simulation, it is essential to select an optimal design plan that maximizes (or minimizes) the performance of the system from limited simulation resources.

To solve this problem, several simulation methods have been proposed, but mainly for a system evaluated with a single performance index. However, most practical systems have a plurality of performance indicators, and it is necessary to select the optimal set of designs, that is, the Pareto set, by considering these performance indicators. For example, when designing a production line system, two (or more) performance indicators such as production cost and quality should be considered, and when designing a semiconductor chip, performance indicators such as power and size of the semiconductor chip should be considered.

The prior art has a problem in that it is difficult to select a Pareto set because only a single performance index is considered.

The present invention was devised to solve the above-described conventional problems, and in a system evaluated by a plurality of performance indicators, an object of the present invention is to select a pareto set, which is a set of optimal designs through probabilistic simulation.

Another object is to improve the accuracy and reliability of the finally selected Pareto set by efficiently allocating limited simulation resources for each design.

In order to solve the above problems, the method for selecting a pareto set according to an embodiment of the present invention is a method of selecting a pareto set based on at least two or more performance indicators by a computer processor. A first simulation resource allocation step of deriving statistical information of each of the design proposals for a simulation result by allocating and performing simulation for each design scheme using the allocated initial simulation resource; A second simulation resource allocation step of calculating the uncertainty value of the design using the statistical information and updating the statistical information by additionally allocating an extra simulation resource corresponding to the uncertainty value; And presenting a design plan included in the Pareto set through the updated statistical information. Including, the second simulation resource allocation step may be repeatedly performed until all of the extra simulation resources are consumed.

The present invention can increase the efficiency of the system design by efficiently selecting the optimal set of Pareto sets through probabilistic simulation for a system evaluated by a plurality of performance indicators.

In addition, the present invention can improve the accuracy and reliability of the selected Pareto set by efficiently using limited simulation resources.

The accompanying drawings, which are included as part of the detailed description to aid understanding of the present invention, provide embodiments of the present invention and describe the technical spirit of the present invention together with the detailed description.
1 is a view for explaining by showing that the design included in the Pareto set among the design of the system evaluated by a plurality of performance indicators.
2 is a flow chart for explaining a method of selecting a Pareto set according to an embodiment of the present invention.
FIG. 3 is a flow chart for explaining in more detail some of the steps of the Pareto set selection method of FIG. 1.
4 to 9 are diagrams for explaining step-by-step a method of selecting a pareto set according to an embodiment of the present invention.

The present invention can be applied to various transformations and can have various embodiments. Hereinafter, specific embodiments will be described in detail with reference to the accompanying drawings.

The following examples are provided to aid in a comprehensive understanding of the methods, devices and / or systems described herein. However, this is only an example and the present invention is not limited thereto.

In describing the embodiments of the present invention, when it is determined that a detailed description of known technology related to the present invention may unnecessarily obscure the subject matter of the present invention, the detailed description will be omitted. In addition, terms to be described later are terms defined in consideration of functions in the present invention, which may vary according to a user's or operator's intention or practice. Therefore, the definition should be made based on the contents throughout this specification. The terminology used in the detailed description is only for describing embodiments of the present invention and should not be limiting. Unless expressly used otherwise, a singular form includes a plural form. In this description, expressions such as “comprising” or “equipment” are intended to indicate certain characteristics, numbers, steps, actions, elements, parts or combinations thereof, and one or more other than described. It should not be interpreted to exclude the presence or possibility of other characteristics, numbers, steps, actions, elements, or parts or combinations thereof.

Further, terms such as first and second may be used to describe various components, but the components are not limited by the terms, and the terms are used to distinguish one component from other components. Used only.

The method for selecting a pareto set according to an embodiment of the present invention is executed in a hardware stage including a memory and a processor. In the following, since the main subject matter of the present invention may be obscured, a description of a hardware stage such as a memory and a processor will be omitted, and a framework of a probabilistic simulation method for selecting a Pareto set operating on a hardware stage will be mainly described. Shall be

Therefore, even if there is no separate description below, the pareto set selection method should be understood as a configuration that is executed and processed on a computer processor of a computing device.

1 is a view for explaining by showing that the design proposal included in the Pareto set among a plurality of design proposals.

The method of selecting a pareto set of the present invention provides a method of selecting a pareto based on at least two performance indicators by a computer processor.

Referring to FIG. 1, it can be seen that a plurality of design proposals are displayed based on actual performance values (μ _{i1 and} μ _i2 ) for two performance indicators. In addition, it can be seen that some of the plurality of designs are included in the Pareto set (S).

The actual performance values of each design (μ _i1, μ _i2 ) may have any value measured by each performance index, and the position on the graph may vary according to the corresponding value. 1 illustrates the results of the simulation performed on the basis of two performance indicators, but the method of selecting a pareto according to an embodiment of the present invention is not limited thereto, and may be performed on three or more performance indicators.

Here, the performance index may refer to a measurement criterion for performance values that are allocated through the allocated simulation resources and allocated simulation resources in an arbitrary design.

The method of selecting a pareto set considering a plurality of performance indicators can be applied to various fields in which a system must be designed by evaluating performance through stochastic simulation. For example, in the MILES (Multiple Integrated Laser Engagement System) for training and simulation of combat situations, finding the optimal design for the width of the laser beam attached to the simulated firearm and the position or detection range of the sensor attached to the clothing. Can be applied. In addition, when designing a semiconductor chip, it can be applied to find the optimal design plan considering power and area. On the other hand, it can be applied to improve the performance of the existing PBS (Poupulation-based search) algorithm (genetic algorithm, particle swarm optimization, etc.) applied in various fields.

In order to define the Pareto set (S) among design proposals, a definition of the dominance relationship is required. Lead relationship is, if the following equation with respect to the two designs x _i and x _j, x _i represents a relationship that is defined in the lead than x _j.

<Equation 1>

Where μ _io -is the actual performance value of the performance index o for the design x _i . That is, for all performance indicators, if the performance value of design x _i is better than or equal to the performance value of x _j , it is defined that design x _i is superior to design x _j . In this case, it is assumed that the lower the performance value, the better the performance.

Referring to FIG. 1, other design proposals may be classified according to areas B1, B2, B3, and B4 based on actual performance values for each performance indicator of design A. In the example of FIG. 1, design A is defined as being superior to designs in area B2.

On the other hand, it is defined as one to <Formula 2> and not in the at least one performance of designs x _i with respect to a performance index if the values are not as good as the performance value for the designs designs x _j x _i x _j as a superior designs. In the example of FIG. 1, design A is defined as not predominant to the designs in areas B1, B3, and B4.

<Equation 2>

And when to <Formula 3> is x _i as that does not superior to x _j, x _j also not superior to x _i to, x _j and x _i is defined that are independent (indifferent) to each other. In the example of FIG. 1, design A is defined as irrelevant to design proposals in areas B1 and B3.

<Equation 3>

Based on the above definitions, the pareto set S based on the actual performance value may be defined as <Equation 4>.

<Equation 4>

Here, the Pareto set S is a set of non-dominated dsigns in which all other designs do not prevail. The design proposals included in Set S have an independent relationship. In addition, designs not included in the set S have an advantage over at least one design set included in the set S.

_io을 바탕으로 추정된 파레토 집합(S_est)을 선택해야 한다. 즉, 추정 성능치

_io는 설계안 x_i에 대해 시뮬레이션을 수행한 결과로 얻은 표본들의 표본 평균값을 의미한다. Since the actual performance value μ _io cannot be obtained based on limited simulation resources, the estimated performance value of the above performance value obtained through limited iteration simulation

_The estimated Pareto set (S _est ) should be selected based on _io . That is, the estimated performance value

_io denotes the sample mean value of the samples obtained as a result of simulation on the design x _i .

<Equation 5>

는 제한된 반복 시뮬레이션을 통해 얻은 추정 성능치를 통한 x_i와 x_j간의 우위 관계를 설명하기 위한 기호이다. 추정 성능치가 포함하고 있는 오류로 인해서 추정되는 파레토 집합(S_est)과 실제 파레토 집합(S) 은 서로 일치하지 않을 수 있다. 따라서, 추정되는 S_est에 대한 정확성은 하기의 <수식 6>과 같이 표현될 수 있다.here

Is a symbol to explain the superiority relationship between x _i and x _j through estimated performance values obtained through limited iteration simulation. The estimated Pareto set (S _est ) and the actual Pareto set (S) may not match each other due to an error included in the estimated performance value. Accordingly, the accuracy for the estimated S _est can be expressed as <Equation 6> below.

<Equation 6>

The more simulation resources, the more simulations are repeated for each design, and the more accurate estimation performance is calculated, thereby increasing the accuracy of S _est . However, since the available simulation resources are limited, it is important to increase the accuracy of the Pareto set S _est calculated and selected within the limited simulation resources.

Hereinafter, a method of more accurately selecting a design scheme included in a Pareto set according to an embodiment of the present invention will be described in detail with reference to the drawings.

2 is a flow chart for explaining a method of selecting a Pareto set according to an embodiment of the present invention. Referring to FIG. 2, in the method for selecting a pareto set according to the present invention, in the method of selecting pareto based on at least two or more performance indicators by a computer processor, a first simulation resource allocation step (S100) and a second simulation resource allocation step (S200) and presenting a design plan included in the Pareto set (S300).

Prior to the first simulation resource allocation step (S100), data on design targets for simulation may be input and stored in a memory used by a processor that processes the simulation.

The first simulation resource allocation step (S100) is a step of deriving statistical information for each of the design proposals for a simulation result by allocating initial simulation resources for a plurality of design proposals and performing simulation for each design scheme using the allocated simulation resources. Can be

The computer processor may allocate at least some of the total simulation resources available in the first simulation resource allocation step S100 as initial simulation resources for each design. In addition, simulation can be performed for each design as much as the initial simulation resource allocated for each design. Through this simulation, the first statistical information for each design can be calculated.

Since there is no simulation result for each design, the first simulation resource in the first simulation resource allocation step S100 may be allocated equally for each design. The value of the initial simulation resource set in the first simulation resource allocation step S100 may be variously changed according to the designer's intention or simulation environment and the size of the available simulation resource.

The computer processor may calculate statistical information based on the results of the simulation for each design in the first simulation resource allocation step (S100) and the second simulation resource allocation step (S200), which will be described below. Here, the statistical information may include an estimated performance value, a sample variance, and a standard error for the performance index calculated as a result of simulation for the design.

Since the computer processor can perform a plurality of simulations for each design, a plurality of simulation results can be generated. Statistical information means information about a plurality of simulation results.

A sample average of performance values derived from a plurality of simulation results generated by the above method may be defined as an estimated performance value of the statistical information. Then, the variance values for a plurality of simulation results may be defined as sample variances of the statistical information. In addition, standard error values for a plurality of simulation results may be defined as standard errors of the statistical information.

The standard error is an error value for the sample mean described above, and the number decreases as the number of simulations increases. The method for selecting a pareto set according to the present invention can improve the accuracy of a selected pareto set within a limited simulation resource by utilizing this standard error value when calculating the uncertainty for allocating additional simulation resources. More specifically, the standard error may mean a value obtained by dividing the square root of the sample variance by the square root of the number of simulations N _i performed for the design.

Even if simulation is performed for one design plan, if there are multiple performance indexes, statistical information including estimated performance values, sample variances, and standard errors for each performance index can be calculated for each design plan. The calculated statistical information is stored prior to processing of the next operation.

When the computer processor calculates the initial statistical information for each design through the first simulation resource allocation step (S100), the second simulation resource allocation step for more accurately updating the statistical information by allocating additional simulation resources ( S200).

The second simulation resource allocation step S200 may be a step of calculating the uncertainty value of the design using the statistical information calculated through the first simulation resource allocation step S100. In addition, the second simulation resource allocation step (S200) may be a step of updating statistical information by performing simulation by additionally allocating simulation resources corresponding to the uncertainty value.

Thereafter, the computer processor may select and present a design plan included in the Pareto set from the estimated performance value of statistical information of each design plan that has been finally updated (S300).

FIG. 3 is a flow chart for explaining in more detail some of the steps of the Pareto set selection method of FIG. 1. Referring to Figure 3, the second simulation resource allocation step (S200) is a step of calculating an uncertainty value using statistical information (S210), additionally allocating a simulation resource corresponding to the uncertainty value (S220) and additionally It may include the step (S230) of updating the statistical information by performing the simulation for each design in response to the allocated simulation resource.

Also, the second simulation resource allocation step S200 may be repeatedly performed until all the simulation resources are consumed. Here, the computer processor may calculate the uncertainty value of the design using statistical information that is updated each time the second simulation resource allocation step S200 is performed. The recalculated uncertainty value can be used again to update statistical information in the design.

Here, the uncertainty value is a measure of whether the estimated performance values, sample variance, and standard error included in the statistical information of the design can be sufficient evidence to verify that the design is included or not included in the Pareto set (S). it means. In other words, the uncertainty value refers to the statistical uncertainty of selection for the design plan based on the calculated statistical information.

According to the present invention, simulation resources allocated for each design scheme may be changed using these uncertainty values in the process of additionally allocating simulation resources. That is, it is possible to allocate more simulation resources to a design having a higher value of uncertainty to prove a decision according to the uncertainty value. Through this, limited simulation resources can be used more efficiently. In addition, it is possible to improve the accuracy and reliability of the finally selected Pareto set.

Hereinafter, a process for calculating an uncertainty value as a criterion for allocating additional simulation resources will be described.

1) In the case of design proposal x _d that does not belong to the estimated Pareto set S _est

The uncertainty value ω _d of the design x _d can be calculated as follows.

<Equation 7>

x_d를 의미한다.Here, the median ε _{i (1), d} , ε _{i (2), d} , ..., ε _{i (l), d} are the median values calculated through <Equation 9> described below, and the subscript i is S Among the designs belonging to _est , these are presumed to have a superior relationship with this design x _d . That is, x _i ∈S _est : x _i

It means x _d .

2) In the case of design plan x _n belonging to the estimated Pareto set S _est

The uncertainty value ω _n of x _n can be calculated as follows.

<Equation 8>

Here, the median values ε _{n, i (1)} , ε _{n, i (2)} , ..., ε _{n, i (a)} are the median values calculated through <Equation 9> described below, and the subscript i is S Among the designs that do not belong to _est , it means the design that has the highest likelihood of being related to design x _n . That is, x _n < x _i means the design that is most likely to be established.

In addition, the median η _{n, j (1)} , η _{n, j (1)} , ..., η _{n, j (b)} are values calculated through Equation 11 described below, and the subscript j is S Among the design _proposals belonging to _est , it means the rest of the design proposals other than itself.

<Equation 9>

<Equation 9> is a formula for calculating the median ε _{i, j} used to calculate the uncertainty calculated in <Equation 7> and <Equation 8>, which is calculated for the two designs x _i and x _j . When it is determined that x _i is superior to x _j according to statistical information, it is used to statistically verify x _i <x _j using hypothesis verification.

In order to verify x _i <x _j , according to <Equation 1>, the relationship of μ _io <μ _jo must be proved for all performance indicators. Each μ _io <μ _jo relationship can be proved true when the p value δ _{io, jo} obtained by each hypothesis test of <Equation 13> below converges to 0. If all p values converge to 0, x _i <x _j proves true. On the other hand, if only one p value converges to 1, x _i <x _j proves to be false because at least one of the μ _io <μ _jo relationships is proven to be false.

Also, when the maximum value among p values is close to 0.5, it is uncertain whether x _i <x _{j is} also true or false because at least one of the above μ _io <μ _jo relationships is true or false. In this context, the value of ε _{i, j} for statistically verifying x _i <x _j is defined as the maximum value among p values for each relationship as shown in <Equation 9>. The definition of p value will be described later.

<Equation 10>

x_j를 가설검증을 활용하여 통계적으로 검증하기 위해 사용된다.<Equation 10> is a formula for calculating γ _{i, j,} which is a value required to obtain the median η _{i, j} used to calculate the uncertainty calculated in Equation 8 above. This is the case according to the previous <Equation 9> the Statistics calculated for the ε _{i, j,} as opposed to the two designs x _i and x _j are calculated in x _i is determined that no relation to the lead than x _j, x _i

It is used to statistically verify x _j using hypothesis verification.

x_j는 배타적인 관계에 있기 때문에 이러한 결과는 불확실성 정의에 대한 일관성을 나타낸다.Where 1-γ _{i, j} is x _i < The uncertainty for x _j is equal to ε _{i, j} and x _i < x _j and x _i

Since x _j is in an exclusive relationship, these results indicate consistency with the definition of uncertainty.

x_j 는 참으로 입증된다. 반면에 모든 성능 지표에 대해서 p 값이 1로 수렴하면 모든 μ_io > μ_jo 관계가 거짓으로 입증되기 때문에 x_i

x_j는 거짓으로 입증된다. 또한 p값들 중에서 최소 값이 0.5에 가까운 경우, 모든 μ_io > μ_jo 관계가 거짓으로 입증되거나 적어도 불확실하다는 것이므로, 따라서 x_i

x_j 또한 참 또는 거짓인지 입증할 수 없으며 불확실하다. 이러한 맥락에서 x_i

x_j 를 통계적으로 검증하기 위한 γ_i,j값은 <수식10>과 같이 각 관계에 대한 p값들 중에서 최소값으로 정의된다.To prove x _i <x _j to be true, the relationship of μ _io > μ _jo must be true for at least one performance indicator. Each μ _io > μ _jo relationship can be proved true when the p value δ _jo , _io obtained by the hypothesis test described below converges to zero. (Where δ _jo and _io are p values to prove μ _io > μ _jo , and p values to prove μ _io <μ _jo satisfy δ _io and _jo as follows: δ _jo , _io = 1-δ _io , _jo ). X _i if at least one p value converges to 0

x _j proves to be true. On the other hand, if p values converge to 1 for all performance metrics, x _i because all μ _io > μ _jo relationships are proven to be false.

x _j proves to be false. Also, if the minimum of the p values is close to 0.5, then all μ _io > μ _jo relationships are proven to be false or at least uncertain, so x _i

x _j also cannot prove whether it is true or false and is uncertain. In this context x _i

The γ _{i, j} value for statistically verifying x _j is defined as a minimum value among p values for each relationship as shown in <Equation 10>.

<Equation 11>

<Equation 11> is a formula for calculating the median η _{i, j} used to calculate the uncertainty calculated in <Equation 8>, and uses the value γ _{i, j} calculated through <Equation 10>. . This is used to statistically verify x _i * x _j when it is determined that the two relationships are irrelevant according to statistical information calculated for the two designs x _i and x _j .

x_j과 x_j

x_i이 모두 참으로 증명되어야 한다. 각각은 <수식 10>에 따라서 각 관계를 증명하는 γ_i,j와 γ_j,i가 모두 0으로 수렴할 때 참으로 증명될 수 있다. 따라서 <수식9>의 경우와 마찬가지로 x_i*x_j를 통계적으로 검증하기 위한 η_i,j값은 <수식 11>과 같이 두 γ_i,j와 γ_j,i값의 최대값으로 정의된다.In order for the two designs to be independent of each other, x _i

x _j and x _j

x _i must all prove true. Each can be proved to be true when γ _{i, j} and γ _{j, i} which prove each relationship according to <Equation 10> converge to 0. Therefore, as in the case of <Equation 9>, the η _{i, j} value for statistically verifying x _i * x _j is defined as the maximum value of two γ _{i, j} and γ _{j, i} values as shown in <Equation 11>.

The uncertainty value ω calculated through <Equation 7> and <Equation 8> is a combination of a minimum value (min) or a maximum value (max) of p-values obtained through simulation statistical information of each design. Is defined. Depending on the nature of the p-value, the uncertainty value means the degree to which the calculated statistical information can be sufficient evidence to determine that the decision is statistically correct. Therefore, the uncertainty value can be used as a criterion for allocating extra simulation resources.

On the other hand, the above-mentioned p value is calculated through a statistical hypothesis test. For example, to select x _i for any performance index (o) is better than selecting x _j (i.e., for performance index o, the performance of x _i is better than the performance of x _j ) and the designs x _i actual performance value of less than μ _io designs x _j actual performance value of μ _jo. That is, μ _io <μ _jo must be satisfied. To prove this, the following statistical hypothesis test can be used. At this time, μ _io <μ _jo to prove is defined by the alternative hypothesis H _A.

<Equation 12>

The p-values δ _io and _jo for this hypothesis test can be calculated as follows.

<Equation 13>

_io 와

_jo는 상기한 바와 같이 설계안 x_i와 x_j에 대하여 성능 지표 o에 있어서 시뮬레이션 결과 계산된 추정 성능치(표본 평균)을 의미한다.In the above formula, X is an arbitrary variable that follows the t-distribution, and δ _io and _jo are p values. s _io is the square root of the sample variance value of the performance index of the designs o x _i, N _i denotes the number of simulated designs performed for x _i.

_{with io}

_jo means the estimated performance value (sample average) calculated as a result of the simulation in the performance index o for the design plans x _i and x _j as described above.

_io, 표본 분산(s_io)^2 및 표준 오차(s_io / √N_i))가 증명하고자하는 μ_io < μ_jo (대립가설)을 참 또는 거짓으로 증명하는데 충분한 증거가 될 수 있는지에 대한 정도를 나타낸다. 즉 p 값이 0에 수렴하는 경우 계산된 통계 정보는 μ_io < μ_jo를 참으로 증명하기에 충분한 증거가 됨을 의미하며 따라서 μ_io < μ_jo는 계산된 결과에 기반하여 참으로 증명된다. 반면에 p 값이 1에 수렴하는 경우 계산된 통계 정보는 μ_io < μ_jo를 거짓으로 증명하기에 충분한 증거가 됨을 의미하며 따라서 μ_io < μ_jo는 계산된된 결과에 기반하여 거짓으로 증명된다 (즉 μ_io > μ_jo이 참으로 증명됨). 반면에 p값이 0.5에 가까운 경우, 계산된 통계 정보가 μ_io < μ_jo를 증명하는 충분한 증거가 되지 못함을 의미하므로 따라서 μ_io < μ_jo는 계산된 통계 정보를 기반하여 참 또는 거짓으로 증명될 수 없다.p value is calculated statistical information of each design (estimated performance value)

_io , whether the sample variance (s _io ) ^ 2 and standard error (s _io / √N _i )) can be sufficient to prove true or false to the μ _io <μ _jo (alternative hypothesis) you want to prove. Degree. That is, when the p value converges to 0, the calculated statistical information is sufficient to prove μ _io <μ _jo , and therefore μ _io <μ _jo is proved true based on the calculated result. On the other hand, if the p value converges to 1 calculated statistical information indicates that there is sufficient evidence to prove the μ _io <μ _jo to be false, and thus μ _io <μ _jo is proved to be false based on the calculated results (Ie μ _io > μ _jo proved true). On the other hand, when the p value is close to 0.5, it means that the calculated statistical information is not sufficient evidence to prove μ _io <μ _jo , so μ _io <μ _jo proves true or false based on the calculated statistical information. Can not be.

Based on the p-values calculated in the above-described way, the intermediate value can be calculated through an equation that is differently applied according to the superiority relationship between design proposals. And finally, the uncertainty value can be calculated using this intermediate value.

The step of allocating additional simulation resources (S220) may be a step of additionally allocating extra simulation resources through the uncertainty value calculated in step S210 of calculating the uncertainty value. Hereinafter, an allocation policy for allocating additional resources to maximize the accuracy of the selected Pareto set based on the calculated uncertainty value will be described.

<Formula 7> can be calculated on the basis of the <Equation 8> the Statistics are calculated uncertainty values of the designs included in the designs and S _est that is not included in S _est through. At this time, the calculated uncertainty value does not exceed 0.5. Therefore, this uncertainty value refers to the degree to which the calculated statistical information is sufficient evidence to prove the choice for this design.

For example, the uncertainty calculated to be in the selected designs x _n <Formula 8> about contained in S _est cases close to zero, the statistics calculated for the designs x _n enough to prove the selection for the designs x _n It means to be evidence. That is, it is correct to select the design plan x _n as being included in the Pareto set.

However, if the value of the uncertainty designs x _n is close to 0.5, which means that the calculated statistics so far can not be an important proof of the choice of x _n designs designs for x _n. That is, it is an uncertain choice to select the design plan x _n as being included in the Pareto set.

In order to maximize the accuracy of S _est (i.e., the selected S _est is equal to S), the calculated uncertainty values for each design should all converge to zero. Therefore, it is necessary to update insufficient statistical information by allocating more simulation resources in a design with a high uncertainty value, that is, a design with an uncertainty value close to 0.5. And through this, the uncertainty value of the design should be lowered. As a result, given a limited number of simulation resources, it is necessary to allocate more simulation resources in a design with a relatively high uncertainty value in order to maximize the accuracy of S _est based on the calculated uncertainty value.

The allocation policy for maximizing the accuracy of S _est based on the uncertainty value is defined as follows.

<Equation 14>

Here, Δ is the number of simulation resources to be allocated, ω _i is the calculated uncertainty value for each design, and a _i is the number of simulation resources each design is allocated according to its calculated uncertainty value. . That is, the Δ refers to the size of the simulation resource allocated in the second simulation resource allocation step. And k means the number of all designs. Here, the size of the simulation resource allocated in the second simulation resource allocation step may be set differently according to the intention of the simulation designer, the simulation environment, or the size of available simulation resources.

4 to 9 are diagrams illustrating the method of selecting a pareto set according to an embodiment of the present invention step by step. 4 to 9 are diagrams for explaining the first simulation resource allocation step (S100) and the second simulation resource allocation step (S200) described above through a graph.

In FIG. 4 to FIG. 9, five design proposals (x ₁ , x ₂ , x ₃ , x ₄ , x ₅ ) are input, and corresponding design schemes (x ₁ , x ₂ , x ₃ , x ₄ , x ₅ ) There is a simulation resource capable of performing 1000 simulations, and in the first simulation resource allocation step (S100), 20 simulation resources (n ₀ ) are allocated for each design plan, and thereafter, until all of the extra simulation resources are consumed. In the repeated second simulation resource allocation step (S200), it is assumed that a total of 10 simulation resources Δ are allocated to the design proposals.

4 and 5 may represent a process in which the computer processor performs the first simulation resource allocation step (S100) for each of the design schemes (x ₁ , x ₂ , x ₃ , x ₄ , x ₅ ). 4, the designs _{(x 1, x 2, x} 3, x 4, x 5) to the initial simulation resources each designs _{(x 1, x 2, x} 3, x 4, x 5) with respect to the It can be seen that 20 are allocated. Thereafter, referring to FIG. 5, simulation is performed as much as a simulation resource for each design, and as a result, statistical information for each performance index is calculated, and estimated performance values and sample variances are calculated based on the statistical information. You can.

Thereafter, the second simulation resource allocation step S200 may be performed by the computer processor. 6 is a view showing a state in which the uncertainty value ω _i for each design is calculated through the step S210 of calculating the uncertainty value in the second simulation resource allocation step S200.

7 is a diagram for the step (S220) of allocating a portion (Δ) of the extra simulation resource corresponding to the calculated uncertainty value (ω _i ). Referring to FIG. 7, it can be seen that more simulation resources are allocated for a design (x ₁ , x ₂ ) having a higher uncertainty value (ω _i ).

8 is a diagram for a step (S230) of performing simulation for each design through the allocated simulation resource and updating the statistical information after the spare simulation resource is allocated.

Each step (S210, S220, S230) shown in FIGS. 6 to 8 may be repeated until all of the extra simulation resources are consumed, and through this iteration, statistical information of each design is continuously updated.

FIG. 9 is a diagram showing statistical information of each design that has been finally updated after all simulation resources are consumed through the second simulation resource allocation step (S200). Referring to FIG. 9, a design plan x ₁ and a design plan x ₂ may be selected as a design plan included in the estimated Pareto set S _est .

4 to 9 are diagrams for explaining a method of selecting a pareto set according to an embodiment of the present invention based on two performance indicators. The present invention can be applied to a simulation environment having two or more performance indicators as well as two or more performance indicators as shown in FIGS. 4 to 9.

The above description is merely illustrative of the technical idea of the present invention, and those skilled in the art to which the present invention pertains may make various modifications and variations without departing from the essential characteristics of the present invention. Therefore, the embodiments described in the present invention are not intended to limit the technical spirit of the present invention, but to explain them, and are not limited to these embodiments. The scope of protection of the present invention should be interpreted by the following claims, and all technical spirits within the scope equivalent thereto should be interpreted as being included in the scope of the present invention.

S100: First simulation resource allocation step
S200: second simulation resource allocation step
S300: Step of presenting a design plan included in the Pareto set

Claims (11)

A method for selecting a Pareto set based on at least two performance indicators by a computer processor,
A first simulation for deriving statistical information of each of the design proposals for a simulation result by allocating the initial simulation resources for a plurality of design schemes identically, and performing simulation for each of the design schemes using the allocated initial simulation resource Resource allocation step;
A second simulation resource allocation for updating the statistical information by calculating the uncertainty value of each of the designs using the statistical information and additionally allocating an extra simulation resource corresponding to the calculated uncertainty value to each of the designs. step; And
Presenting a design plan included in the pareto set through the updated statistical information; Including,
The second simulation resource allocation step is repeatedly performed until all of the extra simulation resources are consumed. delete The method of claim 1, wherein the statistical information
Pareto set selection method comprising the estimated performance value, sample variance, and standard error for the performance index calculated as a result of simulation for the design. According to claim 1, The second simulation resource allocation step
The step of calculating the uncertainty values of the designs using the statistical information that is repeatedly updated. According to claim 1, The second simulation resource allocation step
Calculating uncertainty values of the designs using the statistical information;
Additionally allocating the extra simulation resource corresponding to the uncertainty value; And
Updating the statistical information by performing simulation for each of the design plans in response to an additionally allocated simulation resource; Pareto set selection method comprising a. The method of claim 5, wherein the step of allocating the extra simulation resource is
The pareto set selection method, characterized in that the step of allocating the available simulation resources based on the uncertainty value for each of the design proposals. The method of claim 5, wherein the step of allocating the simulation resource is
A method of selecting a pareto set, characterized in that, for each iteration, the same amount of the simulation resource is allocated for each of the design proposals. The method of claim 5, wherein the step of calculating the uncertainty value is
Calculated through an intermediate value calculated using the p value of an arbitrary design value, the p value

X is an arbitrary variable that follows the t-distribution, δ _io , _jo are p values, and s _io is a sample of estimated performance values calculated after performing a simulation on the performance index o of design x _i Means the square root value of the variance, N _i is the number of iterations of the simulation for the design x _i ,

_{with io}

_jo is a pareto set selection method characterized in that it means the estimated performance values for the design results x _i and x _j in the simulation result. The method of claim 8,
If it does not, the Pareto set of estimates include the arbitrary designs x _d, uncertainty values of the arbitrary designs x _d is

Is calculated through,
The ε _{i (1), d} , ε _{i (2), d} , ..., ε _{i (l), d} are the calculated values based on the p value for any other design of the arbitrary design x _d Method of selecting a set of Pareto, characterized in that the median. The method of claim 8,
If the Pareto set of estimates include the arbitrary designs x _n, uncertainty values of the arbitrary designs x _n is

Is calculated through,
Ε _{n, i (1)} , ε _{n, i (2)} , ..., ε _{n, i (a)} and η _{n, j (1)} , η _{n, j (1)} , ..., η _{n, j (b)} is a pareto set selection method, characterized in that it is the intermediate value calculated based on p values for other designs x _j of the arbitrary designs x _n . The method of claim 8, wherein the step of allocating the simulation resource further
The simulation resource allocated for any design x _i

Is calculated by,
The α _i refers to the number of simulation resources each design proposal is allocated according to its calculated uncertainty value, the Δ means the size of the simulation resources allocated in the second simulation resource allocation step, and ω _i is Pareto set selection method, characterized in that the uncertainty value of the arbitrary design proposal x _i , the k means the number of all the design proposals.

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