CN116701830B - Pareto front edge solution optimization method based on fuzzy rule and stability reasoning control - Google Patents

Pareto front edge solution optimization method based on fuzzy rule and stability reasoning control Download PDF

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CN116701830B
CN116701830B CN202310463135.0A CN202310463135A CN116701830B CN 116701830 B CN116701830 B CN 116701830B CN 202310463135 A CN202310463135 A CN 202310463135A CN 116701830 B CN116701830 B CN 116701830B
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CN116701830A (en
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黄晋
孟天闯
李惠乾
李星宇
张博维
郝建平
王昭清
许宇航
李佳幸
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Tsinghua University
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Abstract

The application relates to a pareto front edge solution optimization method based on fuzzy rules and stability reasoning control. The method comprises the following steps: the method and the device can realize automatic selection of the pareto optimal solution, the selected pareto optimal solution is accurate, the method and the device can be widely applied to the multi-objective optimization problem of generating the pareto front edge solution set, and the reasoning decision intellectualization and calculation efficiency are improved.

Description

Pareto front edge solution optimization method based on fuzzy rule and stability reasoning control
Technical Field
The application relates to the technical field of multi-objective optimization, in particular to a pareto front edge solution optimization method based on fuzzy rules and stability reasoning control.
Background
In the actual multi-objective optimization problem, the solution to the multi-objective optimization problem is not a global optimal solution, but a plurality of pareto front solutions are obtained, and the selection of the better pareto front solution in the solution set of the pareto front solution is usually manually selected, but the manual selection easily causes the inaccuracy of the selected better pareto front solution.
Disclosure of Invention
Based on this, it is necessary to provide a pareto front edge solution optimization method based on fuzzy rule and stability reasoning control, which can select a more accurate objective function.
In a first aspect, the present application provides a pareto front solution preference method. The method comprises the following steps:
respectively acquiring a function value of the first objective function and a function value of the second objective function;
Determining a steady-state solution according to the function value of the first objective function and the function value of the second objective function;
and selecting a target pareto optimal solution from the feasible solution sets of the first objective function and the second objective function according to the steady-state solution.
In one embodiment, determining the steady-state solution according to the function value of the first objective function and the function value of the second objective function includes:
And inputting the function value of the first objective function and the function value of the second objective function into a pre-established fuzzy inference system to obtain a steady-state solution output by the fuzzy inference system.
In one embodiment, the obtaining the steady-state solution output by the fuzzy inference system according to inputting the function value of the first objective function and the function value of the second objective function into the pre-established fuzzy inference system includes:
The function value of the first objective function and the function value of the second objective function are input into a fuzzy reasoning system, the fuzzy reasoning system determines a gravity center real value and a membership area according to a preset membership function and a fuzzy rule base, and a steady-state solution is output according to the gravity center real value and the membership area; the gravity center real value is the gravity center real value of the membership function of the decision variable, and the membership area is the membership area corresponding to the membership function of the decision variable.
In one embodiment, the membership functions include a first membership function corresponding to the first objective function, a second membership function corresponding to the second objective function, and a third membership function corresponding to the decision variable; determining a gravity center real value and a membership area according to a pre-established membership function and a fuzzy rule base, wherein the method comprises the following steps:
according to the first membership function and the second membership function, membership corresponding to the first objective function and membership corresponding to the second objective function are obtained respectively;
determining a matching degree according to the membership degree corresponding to the first objective function and the membership degree corresponding to the second objective function;
And determining the gravity center real value and the membership area according to the matching degree, the third membership function and the fuzzy rule base.
In one embodiment, the above establishment procedure according to the fuzzy inference system includes:
determining the change rate of the true value of the ideal reasoning condition statement according to the ideal reasoning condition statement;
And establishing a fuzzy inference system according to the change rate of the previous sentence of the ideal inference condition sentence and the change rate of the true value of the ideal inference condition sentence.
In one embodiment, the method further comprises:
And determining a standby objective function from the first objective function and the second objective function according to the target pareto optimal solution.
In one embodiment, the method further comprises:
acquiring a plurality of pareto optimal solutions of the multidimensional target optimization problem;
and carrying out normalization processing on the plurality of pareto optimal solutions to obtain a feasible solution set.
In a second aspect, the application also provides a pareto front edge solution optimization device. The device comprises:
The function value determining module is used for respectively acquiring the function value of the first objective function and the function value of the second objective function;
The steady-state solution determining module is used for determining a steady-state solution according to the function value of the first objective function and the function value of the second objective function;
and the optimal solution selection module is used for selecting a target pareto optimal solution from the feasible solution sets of the first objective function and the second objective function according to the steady-state solution.
In a third aspect, the present application also provides a computer device. The computer device comprises a memory storing a computer program and a processor implementing the steps according to the first aspect when the processor executes the computer program.
In a fourth aspect, the present application also provides a computer-readable storage medium. The computer readable storage medium as described above, having stored thereon a computer program which, when executed by a processor, carries out the steps as described in the first aspect.
In a fifth aspect, the present application also provides a computer program product comprising a computer program, the computer program product storing a computer program which, when executed by a processor, carries out the steps as described in the first aspect.
According to the pareto front solution optimizing method based on fuzzy rules and stability reasoning control, the function value of the first objective function and the function value of the second objective function are respectively obtained, then the steady state solution is determined according to the function value of the first objective function and the function value of the second objective function, and finally the objective function is determined according to the feasible solution set and the steady state solution of the first objective function and the second objective function which are obtained in advance.
Drawings
FIG. 1 is a flow diagram of a pareto front solution preference method in one embodiment;
FIG. 2 is a graph of membership functions of related variables of a first objective function in one embodiment;
FIG. 3 is a graph of membership functions of related variables of a second objective function in one embodiment;
FIG. 4 is a graph of membership functions for fuzzy sets of decision variables in one embodiment;
FIG. 5 is a flow chart of determining a centroid real value and a membership area in one embodiment;
FIG. 6 is a graph of centroid real values and membership area in one embodiment;
FIG. 7 is a flow diagram illustrating the setup of a fuzzy inference system in another embodiment;
FIG. 8 is a flow chart of another embodiment for obtaining a feasible solution set;
FIG. 9 is a solution set distribution diagram for solving a minimized two-dimensional objective function in one embodiment;
FIG. 10 is a flow chart of a pareto front solution preference method in another embodiment;
FIG. 11 is a block diagram of a pareto front solution preference device in one embodiment;
Fig. 12 is an internal structural diagram of a computer device in one embodiment.
Detailed Description
The present application will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present application more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the application.
In the practical multi-objective selection problem, the multi-dimensional objective function represents a multi-aspect trade-off, for example, if in the multi-objective selection problem of minimizing the two-dimensional objective function [ J 1,J2 ], the objective function J 1 represents the system performance evaluation, and the objective function J 2 represents the system cost evaluation, which are balanced with each other, because the high performance and the low cost are often difficult to be achieved, at present, the objective function in the final practical application is usually manually selected, but the manually selected objective function is easy to be inaccurate.
As shown in fig. 1, a pareto front edge solution preferred method is provided, and the embodiment of the present application is applied to a terminal for illustration by using the method, it can be understood that the method can also be applied to a server, and can also be applied to a system including the terminal and the server, and implemented through interaction between the terminal and the server. In this embodiment, the method includes the steps of:
Step 201, obtaining a function value of the first objective function and a function value of the second objective function respectively.
The first objective function and the second objective function refer to multidimensional functions acquired by the terminal, and the first objective function and the second objective function include, but are not limited to, multidimensional functions or one-dimensional functions.
When selecting the objective function, the user starts a selection program or inputs a selection instruction. The terminal respectively acquires a first objective function and a second objective function according to a selection program or a selection instruction, and processes the first objective function and the second objective function to obtain corresponding function values.
Illustratively, the terminal obtains a first objective function J 1 and a second objective function J 2, resulting in a function value of the first objective function J 1 =2.5 and a function value of the second objective function J 2 =8.
Step 202, determining a steady-state solution according to the function value of the first objective function and the function value of the second objective function.
The steady state solution accords with the Lyapunov stability, the steady state solution can control the fuzzy reasoning system to be in a state that the true value is not increased and the steady state point is reached, and particularly, the fuzzy reasoning system refers to a system with fuzzy information processing capability based on a fuzzy set theory and fuzzy reasoning.
And the terminal processes the acquired function value of the first objective function and the second objective function to obtain a steady-state solution which can enable the true value of the fuzzy inference system to be non-increased and reach a steady-state point.
Illustratively, if the function value j1 of the first objective function input to the fuzzy inference system=2.5 and the function value j2 of the second objective function=8, the steady state solution of the fuzzy inference system output is.
And 203, selecting a target pareto optimal solution from feasible solution sets of the first objective function and the second objective function according to the steady-state solution.
The target pareto optimal solution is obtained according to an optimal solution on the pareto front, and the target pareto optimal solution can be determined through a position relation between a steady-state solution and a feasible solution, wherein the position relation can include but is not limited to an angle and a coordinate.
And determining the nearest feasible solution according to the steady-state solution output by the fuzzy reasoning system and the position relation between the steady-state solution and the feasible solution, and enabling the feasible solution to be the target pareto optimal solution.
For example, if the steady-state solution is Φ opt =4.71°, according to the acquired azimuth angles corresponding to the respective feasible solutions, the corresponding feasible solution of the nearest azimuth angle is X2, the target pareto optimal solution is
In the above embodiment, the function value of the first objective function and the function value of the second objective function are respectively obtained, then a steady-state solution is determined according to the function value of the first objective function and the function value of the second objective function, and finally an objective pareto optimal solution is determined according to a feasible solution set and a steady-state solution of the first objective function and the second objective function, which are obtained in advance. The embodiment of the application can realize automatic selection of the optimal solution, and the selected target pareto optimal solution is more accurate.
In one embodiment, the step of determining the steady-state solution according to the function value of the first objective function and the function value of the second objective function may include: and inputting the function value of the first objective function and the function value of the second objective function into a pre-established fuzzy inference system to obtain a steady-state solution output by the fuzzy inference system.
The fuzzy reasoning system is a system with the capability of processing fuzzy information based on the technology of fuzzy set theory, fuzzy reasoning and the like, and can realize complex nonlinear mapping by taking the fuzzy theory as a main calculation tool, and the input and the output of the fuzzy reasoning system are accurate numerical values.
The steady-state solution includes, but is not limited to, angles and coordinates, and the membership function of the steady-state solution is a bijective function, i.e., the steady-state solution corresponds to only one different membership function value in a definition domain range.
The terminal inputs the acquired function values of the first objective function and the second objective function into a fuzzy reasoning system, and the fuzzy reasoning system performs fuzzy reasoning on the function values of the first objective function and the second objective function to obtain a steady-state solution with an accurate value.
For example, if the obtained function value of the first objective function is j 1 =2.5 and the function value of the second objective function is j 2 =8, the function values of the first objective function and the second objective function are input into the fuzzy inference system, so as to obtain a steady state solution Φ ppt =45°.
In the above embodiment, the obtained function values of the first objective function and the second objective function are input to the fuzzy inference system, and the fuzzy inference system performs fuzzy inference on the function values to obtain a steady-state solution with an accurate value. According to the embodiment of the application, an accurate steady-state solution is obtained through fuzzy reasoning, and in solving the multi-objective optimization problem, the calculation efficiency can be improved.
In one embodiment, the step of obtaining the steady-state solution output by the fuzzy inference system according to the step of inputting the function value of the first objective function and the function value of the second objective function into the pre-established fuzzy inference system may include: the function value of the first objective function and the function value of the second objective function are input into a fuzzy reasoning system, the fuzzy reasoning system determines a gravity center real value and a membership area according to a preset membership function and a fuzzy rule base, and a steady-state solution is output according to the gravity center real value and the membership area; the gravity center real value is the gravity center real value of the membership function of the decision variable, and the membership area is the membership area corresponding to the membership function of the decision variable.
The fuzzy reasoning system needs to perform fuzzy reasoning according to a pre-established fuzzy rule base, the fuzzy rule base is clue with a spare force on the design according to the experience and knowledge of an expert, the fuzzy rule base comprises a plurality of fuzzy rules, and the fuzzy rules are used for defining and describing the experience relation between each dimension target balance and pareto optimal solution selection, namely the relation between the pareto optimal front solution set defining a multi-dimension target function and decision. Such empirical relationships, for example, in a multi-objective optimization problem that considers both performance and cost objective functions, if the budget is sufficient and the performance requirements are high, the pareto optimal solution with better performance evaluation function is more prone to be selected; if the budget is limited and the performance requirements are not very high, the pareto optimal solution with better cost evaluation function is more prone to be selected.
Wherein, the fuzzy rule base R consists of n R if-then rules as shown in formula (1):
each if-then rule r i,i＝1,2,…,nR defines an empirical relationship between M-dimensional targets and pareto optimal solution selection decisions, and its structure generally includes M target-related propositions (p 1,p2,…,pM) and 1 pareto optimal solution selection decision-related propositions (q), as shown in formula (2):
Ifp1andp2and…andpM,thenq....................................(2)
Where p 1,p2...pm describes the requirements or constraints associated with the objective function J i, it can be customized according to the actual task, the specific objective dimensions. For example, if in a design issue, J i measures system performance, P i may be defined as "system performance requirement high/low" or "system tolerance to errors and delays low/high", etc.; if J i measures system cost, P i may be defined as "system budget more/less" or the like. Generally, P i has the form of "J iisFi", J i is a variable related to a target dimension, such as "system performance requirement", "tolerance of system to errors and delays", "system budget", and the like, and F i is a fuzzy set describing J i, such as "very high (multiple)", "relatively high (multiple)", "not high (not low (not too much)", "relatively low (few)", "very low (few)", and the like, and generally, the concepts of similar height, how much, size, and the like can be collectively represented by a set of positive large (PB), median (PM), positive Small (PS), zero (ZO), negative Small (NS), negative Median (NM), negative large (NB), and the like.
In addition, the proposition q describes a decision for selecting the pareto optimal solution, for example, "the cost performance ratio of the selected pareto optimal solution is high/low", and it should be noted that the proposition q needs to be defined with the capability of uniquely distinguishing each solution on the front, taking the definition as an example, it needs to ensure that the cost performance ratio of each point on the front is different (actually, all points on the pareto front of the performance and cost two-dimensional objective function are considered to have different cost performance ratios), and only then, it can be ensured that after we determine the value of the ideal cost performance ratio by reasoning, the value of the ideal cost performance ratio can uniquely determine one point on the front as the finally selected optimal solution. Generally, q also has the form of "phi isG", phi is a variable related to a decision for selecting a pareto optimal solution, such as "cost performance ratio", and the like, G is a fuzzy set describing the decision related variable phi, such as "high", "low", "large", "small", and the like, and is generally also uniformly represented by a set of positive large (PB), positive Small (PS), zero (ZO), negative Small (NS), negative Medium (NM), negative large (NB), and the like.
Thus, as shown in equation (3), each rule is also noted:
Ifj1isF1andj2isF2and…andjMisFM,thenφisG.........(3)
When F i and G adopt 7 sets of descriptions such as positive large (PB), positive Small (PS), zero (ZO), negative Small (NS), negative Medium (NM), negative large (NB), etc., n R＝7M rules can be formulated altogether, and the following is a rule example, as shown in formula (4):
Ifj1isPBandj2isNMand…andjMisZO,thenφisPS.....(4)
The membership function refers to membership of elements to fuzzy sets, and is established according to the fuzzy sets established in advance and actual values of target variables and decision variables, as shown in fig. 2 to 4, and comprises a first target function J 1 for measuring performance, a second target function J 2 for measuring cost and a membership function corresponding to the decision variables. As shown in fig. 2, the horizontal axis of the objective function J 1 represents the related variable J 1, which is used to represent the "system performance requirement level", the range of values is [0, 10], the closer the value is to 10, the higher the system performance requirement level is, i.e. the smaller the corresponding requirement objective function J 1, and the vertical axis represents the membership degree of the related variable to each fuzzy subset. As shown in fig. 3, the horizontal axis of the objective function J 2 represents the related variable J 2, which is used to represent "the system residual budget", and is represented by the value of [0, 10], where the unit is thousand yuan, the larger the value is, the more the system residual budget is, and the vertical axis represents the membership of the related variable to each fuzzy subset. As shown in fig. 4, the horizontal axis of the membership function of the decision variable represents the azimuth angle of each pareto front point relative to the origin after normalization, the value range is [0, 90], the unit is degree (°), and the vertical axis represents the membership degree of the relevant variable to each fuzzy subset.
The gravity center real value of the membership function of the decision variable refers to the gravity center value of a graph formed by the membership function of the decision variable and the transverse axis, and the membership area refers to the membership area intercepted downwards from the membership function of the decision variable.
The terminal inputs the obtained function value of the first objective function, the obtained function value of the second objective function, the obtained membership function corresponding to the first objective function, the obtained membership function of the second objective function and the obtained membership function of the decision variable into a fuzzy reasoning system, and the fuzzy reasoning system determines a gravity center real value and a membership area according to the membership functions and a pre-established fuzzy rule base and obtains and outputs a steady state solution according to the gravity center real value and the membership area.
In the above embodiment, the obtained function value of the first objective function, the obtained function value of the second objective function, the obtained membership function corresponding to the first objective function, the obtained membership function of the second objective function, and the obtained membership function of the decision variable are input into the fuzzy inference system, and the fuzzy inference system determines the real gravity center value and the membership area according to the membership functions and the fuzzy rule library established in advance, and outputs the steady-state solution according to the real gravity center value and the membership area. According to the embodiment of the application, the accurate steady-state solution is obtained by calculating the gravity center real value and the membership area, so that the calculation efficiency and the calculation accuracy are improved.
In one embodiment, the membership functions include a first membership function corresponding to a first objective function, a second membership function corresponding to a second objective function, and a third membership function corresponding to a decision variable; as shown in fig. 5, the step of determining the gravity center real value and the membership area according to the pre-established membership function and the fuzzy rule base may include:
Step 301, obtaining a membership degree corresponding to the first objective function and a membership degree corresponding to the second objective function according to the first membership function and the second membership function, respectively.
Wherein, the membership degree refers to the degree that an element belongs to a fuzzy set.
The terminal obtains a first objective function and a second objective function, obtains a first membership function corresponding to the first objective function and a second membership function corresponding to the second objective function according to the fuzzy set, and obtains membership degrees of the first objective function and the second objective function to the corresponding fuzzy set according to the fuzzy set corresponding to the first membership function and the second membership function in the current fuzzy rule in the fuzzy rule base.
For example, a first objective function J1 and a second objective function J2 are obtained, where the first objective function J1 corresponds to a membership degree of 0.3 to the fuzzy set PB, and the second objective function J2 corresponds to a membership degree of 0.7 to the fuzzy set NB, according to the fuzzy set PB corresponding to the first objective function J1, the fuzzy set NB corresponding to the second objective function J2, the first membership degree function a 1 (x), and the second membership degree function a 2 (x).
Step 302, determining the matching degree according to the membership degree corresponding to the first objective function and the membership degree corresponding to the second objective function.
The matching degree refers to the matching degree of the objective function to rules in the fuzzy rule base.
After the membership degree of the corresponding fuzzy set is found out according to the first membership degree function and the second membership degree function, the membership degree with the minimum numerical value is selected from the membership degree of the first membership degree function and the membership degree of the second membership degree function; and determining the matching degree with the current fuzzy rule in the fuzzy rule base according to the membership degree with the minimum value.
For example, if the current fuzzy rule is r i:Ifj1isPBandj2 isNM, then Φisps, the membership degree μ NM(j2 of the function value of j 1 to the membership degree μ PB(j1)＝0.7,j2 of the fuzzy set PB to the membership degree μ NM(j2 of the fuzzy set nm=0.3, and the membership degree with the smallest value is μ NM(j2) =0.3, then the matching degree of [ j 1,j2 ] to the current fuzzy rule r i is as shown in formula (5):
wi＝min{μPB(j1),μNM(j2)}＝0.3..................................(5)
And step 303, determining a gravity center real value and a membership area according to the matching degree, the third membership function and the fuzzy rule base.
The third membership function refers to a membership function corresponding to the decision variable.
According to the calculated matching degree of the fuzzy rule, the third membership function corresponding to the decision variable and the pre-established fuzzy rule library, as shown in fig. 6, the membership area is "intercepted downwards" from the third membership function according to the matching degree, and the gravity center real value is the gravity center value of the graph (triangle) formed by the third membership function and the transverse axis.
In the above embodiment, the barycenter real value and the membership area are calculated according to the matching degree of the fuzzy rule, the third membership function corresponding to the decision variable and the pre-established fuzzy rule base, and the barycenter real value and the membership area are solved according to the matching degree of the fuzzy rule, the third membership function corresponding to the decision variable and the pre-established fuzzy rule base, and the stable solution can be accurately obtained by calculating the barycenter real value and the membership area, so that the objective function is accurately selected according to the stable solution.
In one embodiment, as shown in fig. 7, the above establishment procedure according to the fuzzy inference system includes:
step 401, determining the change rate of the true value of the ideal reasoning condition statement according to the ideal reasoning condition statement.
Wherein, the ideal inference condition statement refers to an "if-then ideal inference condition statement", as shown in formula (6), which refers to:
If decision related variable phi isH, then decision result is excellent............................. (6)
Wherein the true value of the if-then ideal inference condition statement is represented by fuzzy implication I (x, f (x)).
X is the true value of if front part statement (i.e. "decision related variable phi isH"), which is equal to the membership of phi to fuzzy set H, i.e. x=μ H (phi), where fuzzy set H and the membership function μ H (·) of phi to fuzzy set H need to be specifically defined, membership function μ H (·) needs to be a bijective function, i.e. each Φe [ phi min,φmax ] corresponds to a unique different value of μ H (phi), fuzzy set H can be generally defined as "large/small" near the upper/lower bounds of phi ", etc.
F (x) is the true value of the statement (namely 'better decision result') and is a function about x preset by the fuzzy inference system, and the steady-state point of the dynamic fuzzy inference system designed later is required to be ensured to correspond to the better decision result.
If-then ideal reasoning condition statement truth I (x, f (x)) takes a type Reichenbach of fuzzy implication and simplifies the truth symbol to z, as shown in equation (7), with the following form:
z＝I(x,f(x))＝1-x+xf(x)......................................(7)
As shown in formula (8), the rate of change of the if-then ideal reasoning condition statement truth value is:
and carrying out reasoning selection on the pareto optimal solution by combining the fuzzy rule base, and determining the change rate of the true value of the ideal reasoning condition statement according to the ideal reasoning condition statement.
Illustratively, if the ideal inference condition statement is as shown in formula (9):
if decision related variable phi isH, then decision result is excellent............................. (9)
As shown in formula (10), the rate of change of the true value of the ideal inference condition statement is:
Step 402, a fuzzy inference system is built according to the change rate of the former sentence of the ideal inference condition sentence and the change rate of the true value of the ideal inference condition sentence.
Wherein, the fuzzy inference system is controlled by the fuzzy inference controller, in the inference process, we need to continuously adjust (control) the decision related variable phi, and the variable x=μ H (phi) directly corresponds to phi, so we let the fuzzy inference controller u be the change rate of x, as shown in formula (11):
substituting the fuzzy inference controller u into the true value change rate of if-then ideal inference condition statement to construct a dynamic fuzzy inference system based on fuzzy true value evolution, as shown in formula (12):
for a fuzzy inference system, for a true value z=1-x+xf (x), the predefined f (x) has the following form, as shown in equation (13):
the corresponding true values and their derivatives are shown in equation (14) and equation (15):
Wherein h FR is a variable related to the fuzzy rule base constructed based on empirical knowledge in section 3.2, and the expression of h FR is as shown in formula (16):
Wherein i represents a rule sequence, C i is a real value of a gravity center of a membership function of a decision variable phi in a rule r i, S i is a membership area obtained by "intercepting down" a matching degree w i of a current target dimension related variable [ j 1,j2,…,jM ] to the rule r i from the membership function of the decision variable phi, and a matching degree w i of the current target dimension related variable [ j 1,j2,…,jM ] to the rule r i can be obtained by using a min function, as shown in formula (17):
Explaining the above by taking two-dimensional multi-objective optimization as an example, let the objective function be [ J 1,J2 ], the objective dimension-related variable [ J 1,j2 ], and the fuzzy rule r i as shown in formula (18):
ri:Ifj1isPBand2isNM,thenφisPS.......................(18)
Membership functions μ PB(j1)、μNM(j2) and μ PS (Φ) as shown in fig. 8, the function value of the first objective function j 1 corresponds to the membership μ PB(j1 of the fuzzy set pb=0.7, the function value of the second objective function j 2 corresponds to the membership μ NM(j2 of the fuzzy set nm=0.3, and then the matching degree w i of [ j 1,j2 ] to the rule r i is as shown in formula (19):
wi＝min{μPB(j1),μNM(j2)}＝0.3.............................(19)
As shown in fig. 6, the membership area S i of the matching degree w i "cut down" from the membership function of the decision variable Φ is shown in the gray shaded portion, and C i is the barycenter real value of the graph (triangle) formed by the μ PS (Φ) function and the horizontal axis.
The process obtains the change rate of the former sentence and the change rate of the true value of the ideal reasoning condition sentence according to the ideal reasoning condition sentence, thereby establishing a fuzzy reasoning system.
In the above embodiment, the pareto optimal solution is inferred and selected by combining with the fuzzy rule base, and the change rate of the true value of the ideal inference condition statement, the change rate of the former statement and the change rate of the true value of the ideal inference condition statement are determined according to the ideal inference condition statement, so as to establish the fuzzy inference system. The fuzzy inference system established by the embodiment of the application can accurately output the steady-state solution, thereby improving the selection accuracy of the objective function.
In one embodiment, the method further comprises: and determining a standby objective function from the first objective function and the second objective function according to the target pareto optimal solution.
The target pareto optimal solution is obtained according to the feasible solution set on the pareto front, and the target pareto optimal solution can be determined through the position relation between the steady-state solution and the feasible solution, wherein the position relation can include but is not limited to angles and coordinates.
After finding out the nearest target pareto optimal solution according to the steady-state solution, determining the nearest feasible solution according to the position relation between the steady-state solution and the feasible solution, determining a standby target function according to the target pareto optimal solution, wherein the standby target function can be one or more and depends on the value of the target pareto optimal solution.
For example, the first objective function J1 and the second objective function J2 are obtained, a steady-state solution obtained by solving a feasible solution set of the first objective function J1 and the second objective function J2 for X1:(J1,1,J2,1)＝(0.85,12),x2:(J1,2,J2,2)＝(0.56,15),X3:(J1,3,J2,3)＝(0.36,32),X4:(J1,4,J2,4)＝(0.18,56),X5:(J1,5,J2,5)＝(0.05,75), is Φ opt =45°, a feasible solution X4 obtained by the steady-state solution is the most practical, and then the standby objective function is determined as the first objective function J1 according to the feasible solution X4.
In the above embodiment, the standby objective function is determined from the first objective function and the second objective function according to the target pareto optimal solution, and the embodiment of the application enables the fuzzy inference controller to be always at a steady point through the steady-state solution, thereby improving the stability of the system, and the embodiment of the application selects the objective function according to the target pareto optimal solution, thereby improving the accuracy of selecting the objective function.
In one embodiment, as shown in fig. 8, the embodiment of the present application may further include the following steps:
step 501, obtaining a plurality of pareto optimal solutions of the multidimensional objective optimization problem.
Wherein, pareto optimal solution refers to: if no other solution exists in the variable space that can be better than this solution, then this solution is called pareto optimal solution. The solution x being better than the solution y means that all the objective functions of the solution x are better than the objective functions corresponding to the solution y.
And obtaining a plurality of objective functions according to the multi-dimensional objective selection problem to be solved, and calculating a plurality of pareto optimal solutions according to the objective functions.
Illustratively, to solve the cost optimization problem, the resulting optimized performance objective function J 1 and cost objective function J 2, in turn, result in the pareto optimal solution being ：X1:(J1,1,J2,1)＝(0.85,12),X2:(J1,2,J2,2)＝(0.56,15),X3:(J1,3,J2,3)＝(0.36,32),X4:(J1,4,J2,4)＝(0.18,56),X5:(J1,5,J2,5)＝(0.05,75).
Step 502, performing normalization processing on a plurality of pareto optimal solutions to obtain a feasible solution set.
Wherein, pareto optimal solution refers to: if no other solution exists in the variable space that can be better than this solution, then this solution is called pareto optimal solution. The solution x being better than the solution y means that all the objective functions of the solution x are better than the objective functions corresponding to the solution y. Taking fig. 9 as an example, the solution set obtained after the minimized two-dimensional objective function [ J 1,J2 ] is solved, and the 5 black points (a-E) are pareto optimal solutions, so that the pareto front is formed together, and the solution B and the solution C are better than the solution F, and the solution C and the solution D are better than the solution G, so that the solution F and the solution G are not pareto optimal solutions.
The multi-objective selection problem generally does not have a globally optimal solution with optimal cost and optimal performance, but rather obtains a solution set comprising a plurality of feasible solutions, that is, a plurality of Pareto (Pareto) optimal solutions, which together form a Pareto front.
After solving the multi-objective selection problem to obtain a plurality of feasible solutions on the pareto front, one of the plurality of feasible solutions is required to be selected as a solution for final practical application, the selection process often needs to be considered in combination with practical situations, and how to select the pareto optimal solution which is more suitable for the practical situations and the problem requirements is an important problem.
The normalization can normalize objective functions of all the pareto optimal solutions in each dimension within the range of [0,1], so that each solution is distributed more uniformly, subsequent calculation is facilitated, and the problems that calculation is complicated or solutions are too dense in a certain dimension due to too large difference of orders of magnitude of different objective functions are avoided.
If N points (pareto optimal solution) exist on the pareto front P obtained by solving the M-dimensional target optimization problem, the N points are marked as P= { x 1,x2,…,xN }
Wherein x i, i=1, 2, …, N represents a value having M-dimensional objective function, denoted as x i＝[J1,i,J2,i,…,JM,i.
Where J j,i, j=1, 2, …, M represents the J-th dimensional objective function of point x i.
Recording the maximum value of all pareto optimal solutions on the j-th dimension objective functionAnd minimumAs shown in formulas (20) and (21):
Normalized value formulas (22) and (23) of pareto optimal solution x i show:
for/>
After normalization, all normalized objective function valuesAre all within the range of [0, 1].
And carrying out normalization processing according to the obtained multiple pareto optimal solutions to obtain a feasible solution set in the range of [0,1 ].
Illustratively, normalizing one of the pareto front points results in the following equation (24):
In the embodiment, according to the method, the device and the system for obtaining the pareto optimal solutions of the multidimensional objective optimization problem, the plurality of pareto optimal solutions are normalized to obtain the feasible solution set, and the feasible solution obtained by normalization in the embodiment of the application solves the problem that the plurality of solutions are too dense in a certain dimension, so that the convenience of calculation is improved.
In one embodiment, as shown in fig. 10, the embodiment of the present application further includes the following steps:
step 601, obtaining a function value of the first objective function and a function value of the second objective function respectively.
Step 602, inputting the function value of the first objective function and the function value of the second objective function into a pre-established fuzzy inference system.
And step 603, the fuzzy inference system acquires the membership degree corresponding to the first objective function and the membership degree corresponding to the second objective function according to the first membership degree function and the second membership degree function respectively.
In step 604, the fuzzy inference system determines a matching degree according to the membership degree corresponding to the first objective function and the membership degree corresponding to the second objective function.
And step 605, the fuzzy inference system determines a gravity center real value and a membership area according to the matching degree, the third membership function and the fuzzy rule base.
In step 606, the fuzzy inference system outputs a steady-state solution according to the gravity center real value and the membership area.
Step 607, selecting a target pareto optimal solution from the feasible solution sets of the first objective function and the second objective function according to the steady-state solution.
Step 608, determining a standby objective function from the first objective function and the second objective function according to the target pareto optimal solution.
According to the method, the target pareto optimal solution is determined by inputting the function value of the first target function and the function value of the second target function into a pre-established fuzzy inference system, outputting a steady-state solution, and selecting a target pareto optimal solution from feasible solutions of the first target function and the second target function, and the pareto optimal solution is selected in a reasoning manner by designing the fuzzy inference controller to control the fuzzy inference system to be steady state.
It should be understood that, although the steps in the flowcharts related to the embodiments described above are sequentially shown as indicated by arrows, these steps are not necessarily sequentially performed in the order indicated by the arrows. The steps are not strictly limited to the order of execution unless explicitly recited herein, and the steps may be executed in other orders. Moreover, at least some of the steps in the flowcharts described in the above embodiments may include a plurality of steps or a plurality of stages, which are not necessarily performed at the same time, but may be performed at different times, and the order of the steps or stages is not necessarily performed sequentially, but may be performed alternately or alternately with at least some of the other steps or stages.
Based on the same inventive concept, the embodiment of the application also provides a pareto front solution optimizing device for realizing the above-mentioned pareto front solution optimizing method. The implementation of the solution provided by the device is similar to that described in the above method, so the specific limitations in the embodiments of the one or more pareto front solution preferred devices provided below may be referred to above as limitations on the pareto front solution preferred method, and are not repeated here.
In one embodiment, as shown in fig. 11, there is provided a pareto front solution preference apparatus, comprising:
the function value determining module 701 is configured to obtain a function value of the first objective function and a function value of the second objective function respectively;
A steady-state solution determining module 702, configured to determine a steady-state solution according to the function value of the first objective function and the function value of the second objective function;
the optimal solution selection module 703 is configured to determine an objective function according to a feasible solution set and a steady-state solution of the first objective function and the second objective function, which are acquired in advance.
In one embodiment, the steady-state solution determining module 702 is specifically configured to input the function value of the first objective function and the function value of the second objective function into a pre-established fuzzy inference system, so as to obtain a steady-state solution output by the fuzzy inference system.
In one embodiment, the steady-state solution determining module 702 is specifically configured to input the function value of the first objective function and the function value of the second objective function into the fuzzy inference system, determine, by the fuzzy inference system, a real gravity center value and a membership area according to a membership function and a fuzzy rule base that are established in advance, and output a steady-state solution according to the real gravity center value and the membership area; the gravity center real value is the gravity center real value of the membership function of the decision variable, and the membership area is the membership area corresponding to the membership function of the decision variable.
In one embodiment, the steady state solution determination module 702 further includes:
The membership determination submodule is used for respectively acquiring membership corresponding to the first objective function and membership corresponding to the second objective function according to the first membership function and the second membership function;
the matching degree determination submodule is used for determining matching degree according to the membership degree corresponding to the first objective function and the membership degree corresponding to the second objective function;
and the membership area determination submodule is used for determining a gravity center real value and a membership area according to the matching degree, the third membership function and the fuzzy rule base.
In one embodiment, the apparatus further comprises:
the change rate determining module is used for determining the change rate of the true value of the ideal reasoning condition statement according to the ideal reasoning condition statement;
the system establishment module is used for establishing a fuzzy inference system according to the change rate of the previous sentence of the ideal inference condition sentence and the change rate of the true value of the ideal inference condition sentence.
In one embodiment, the optimal solution selection module 703 includes:
and the standby objective function determining submodule is used for determining a standby objective function from the first objective function and the second objective function according to the target pareto optimal solution.
The above-mentioned pareto front-edge solution optimizing means may be implemented in whole or in part by software, hardware and combinations thereof. The above modules may be embedded in hardware or may be independent of a processor in the computer device, or may be stored in software in a memory in the computer device, so that the processor may call and execute operations corresponding to the above modules.
In one embodiment, a computer device is provided, which may be a terminal or a server, and the internal structure of which may be as shown in fig. 12. The computer device includes a processor, a memory, an Input/Output interface (I/O) and a communication interface. The processor, the memory and the input/output interface are connected through a system bus, and the communication interface is connected to the system bus through the input/output interface. Wherein the processor of the computer device is configured to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, computer programs, and a database. The internal memory provides an environment for the operation of the operating system and computer programs in the non-volatile storage media. The database of the computer device is used for storing objective function related variable data. The input/output interface of the computer device is used to exchange information between the processor and the external device. The communication interface of the computer device is used for communicating with an external terminal through a network connection. The computer program is executed by a processor to implement a pareto front-edge solution preference method.
It will be appreciated by those skilled in the art that the structure shown in FIG. 12 is merely a block diagram of some of the structures associated with the present inventive arrangements and is not limiting of the computer device to which the present inventive arrangements may be applied, and that a particular computer device may include more or fewer components than shown, or may combine some of the components, or have a different arrangement of components.
In one embodiment, a computer device is provided comprising a memory and a processor, the memory having stored therein a computer program, the processor when executing the computer program performing the steps of:
respectively acquiring a function value of the first objective function and a function value of the second objective function;
Determining a steady-state solution according to the function value of the first objective function and the function value of the second objective function;
and selecting a target pareto optimal solution from the feasible solution sets of the first objective function and the second objective function according to the steady-state solution.
In one embodiment, the processor when executing the computer program further performs the steps of:
And inputting the function value of the first objective function and the function value of the second objective function into a pre-established fuzzy inference system to obtain a steady-state solution output by the fuzzy inference system.
In one embodiment, the processor when executing the computer program further performs the steps of:
The function value of the first objective function and the function value of the second objective function are input into a fuzzy reasoning system, the fuzzy reasoning system determines a gravity center real value and a membership area according to a preset membership function and a fuzzy rule base, and a steady-state solution is output according to the gravity center real value and the membership area; the gravity center real value is the gravity center real value of the membership function of the decision variable, and the membership area is the membership area corresponding to the membership function of the decision variable.
In one embodiment, the processor when executing the computer program further performs the steps of:
according to the first membership function and the second membership function, membership corresponding to the first objective function and membership corresponding to the second objective function are obtained respectively;
determining a matching degree according to the membership degree corresponding to the first objective function and the membership degree corresponding to the second objective function;
And determining the gravity center real value and the membership area according to the matching degree, the third membership function and the fuzzy rule base.
In one embodiment, the processor when executing the computer program further performs the steps of:
determining the change rate of the true value of the ideal reasoning condition statement according to the ideal reasoning condition statement;
And establishing a fuzzy inference system according to the change rate of the previous sentence of the ideal inference condition sentence and the change rate of the true value of the ideal inference condition sentence.
In one embodiment, the processor when executing the computer program further performs the steps of:
And determining a standby objective function from the first objective function and the second objective function according to the target pareto optimal solution.
In one embodiment, the processor when executing the computer program further performs the steps of:
acquiring a plurality of pareto optimal solutions of the multidimensional target optimization problem;
and carrying out normalization processing on the plurality of pareto optimal solutions to obtain a feasible solution set.
In one embodiment, a computer readable storage medium is provided having a computer program stored thereon, which when executed by a processor, performs the steps of:
respectively acquiring a function value of the first objective function and a function value of the second objective function;
Determining a steady-state solution according to the function value of the first objective function and the function value of the second objective function;
and selecting a target pareto optimal solution from the feasible solution sets of the first objective function and the second objective function according to the steady-state solution.
In one embodiment, the computer program when executed by the processor further performs the steps of:
And inputting the function value of the first objective function and the function value of the second objective function into a pre-established fuzzy inference system to obtain a steady-state solution output by the fuzzy inference system.
In one embodiment, the computer program when executed by the processor further performs the steps of:
The function value of the first objective function and the function value of the second objective function are input into a fuzzy reasoning system, the fuzzy reasoning system determines a gravity center real value and a membership area according to a preset membership function and a fuzzy rule base, and a steady-state solution is output according to the gravity center real value and the membership area; the gravity center real value is the gravity center real value of the membership function of the decision variable, and the membership area is the membership area corresponding to the membership function of the decision variable.
In one embodiment, the computer program when executed by the processor further performs the steps of:
according to the first membership function and the second membership function, membership corresponding to the first objective function and membership corresponding to the second objective function are obtained respectively;
determining a matching degree according to the membership degree corresponding to the first objective function and the membership degree corresponding to the second objective function;
And determining the gravity center real value and the membership area according to the matching degree, the third membership function and the fuzzy rule base.
In one embodiment, the computer program when executed by the processor further performs the steps of:
determining the change rate of the true value of the ideal reasoning condition statement according to the ideal reasoning condition statement;
And establishing a fuzzy inference system according to the change rate of the previous sentence of the ideal inference condition sentence and the change rate of the true value of the ideal inference condition sentence.
In one embodiment, the computer program when executed by the processor further performs the steps of:
And determining a standby objective function from the first objective function and the second objective function according to the target pareto optimal solution.
In one embodiment, the computer program when executed by the processor further performs the steps of:
acquiring a plurality of pareto optimal solutions of the multidimensional target optimization problem;
and carrying out normalization processing on the plurality of pareto optimal solutions to obtain a feasible solution set.
In one embodiment, a computer program product is provided comprising a computer program which, when executed by a processor, performs the steps of:
respectively acquiring a function value of the first objective function and a function value of the second objective function;
Determining a steady-state solution according to the function value of the first objective function and the function value of the second objective function;
and selecting a target pareto optimal solution from the feasible solution sets of the first objective function and the second objective function according to the steady-state solution.
In one embodiment, the computer program when executed by the processor further performs the steps of:
And inputting the function value of the first objective function and the function value of the second objective function into a pre-established fuzzy inference system to obtain a steady-state solution output by the fuzzy inference system.
In one embodiment, the computer program when executed by the processor further performs the steps of:
The function value of the first objective function and the function value of the second objective function are input into a fuzzy reasoning system, the fuzzy reasoning system determines a gravity center real value and a membership area according to a preset membership function and a fuzzy rule base, and a steady-state solution is output according to the gravity center real value and the membership area; the gravity center real value is the gravity center real value of the membership function of the decision variable, and the membership area is the membership area corresponding to the membership function of the decision variable.
In one embodiment, the computer program when executed by the processor further performs the steps of:
according to the first membership function and the second membership function, membership corresponding to the first objective function and membership corresponding to the second objective function are obtained respectively;
determining a matching degree according to the membership degree corresponding to the first objective function and the membership degree corresponding to the second objective function;
And determining the gravity center real value and the membership area according to the matching degree, the third membership function and the fuzzy rule base.
In one embodiment, the computer program when executed by the processor further performs the steps of:
determining the change rate of the true value of the ideal reasoning condition statement according to the ideal reasoning condition statement;
And establishing a fuzzy inference system according to the change rate of the previous sentence of the ideal inference condition sentence and the change rate of the true value of the ideal inference condition sentence.
In one embodiment, the computer program when executed by the processor further performs the steps of:
And determining a standby objective function from the first objective function and the second objective function according to the target pareto optimal solution.
In one embodiment, the computer program when executed by the processor further performs the steps of:
acquiring a plurality of pareto optimal solutions of the multidimensional target optimization problem;
and carrying out normalization processing on the plurality of pareto optimal solutions to obtain a feasible solution set.
It should be noted that, the user information (including but not limited to user equipment information, user personal information, etc.) and the data (including but not limited to data for analysis, stored data, presented data, etc.) related to the present application are information and data authorized by the user or sufficiently authorized by each party, and the collection, use and processing of the related data need to comply with the related laws and regulations and standards of the related country and region.
Those skilled in the art will appreciate that implementing all or part of the above described methods may be accomplished by way of a computer program stored on a non-transitory computer readable storage medium, which when executed, may comprise the steps of the embodiments of the methods described above. Any reference to memory, database, or other medium used in embodiments provided herein may include at least one of non-volatile and volatile memory. The nonvolatile Memory may include Read-Only Memory (ROM), magnetic tape, floppy disk, flash Memory, optical Memory, high density embedded nonvolatile Memory, resistive random access Memory (ReRAM), magneto-resistive random access Memory (Magnetoresistive Random Access Memory, MRAM), ferroelectric Memory (Ferroelectric Random Access Memory, FRAM), phase change Memory (PHASE CHANGE Memory, PCM), graphene Memory, and the like. Volatile memory can include random access memory (Random Access Memory, RAM) or external cache memory, and the like. By way of illustration, and not limitation, RAM can be in various forms such as static random access memory (Static Random Access Memory, SRAM) or dynamic random access memory (Dynamic Random Access Memory, DRAM), etc. The databases referred to in the embodiments provided herein may include at least one of a relational database and a non-relational database. The non-relational database may include, but is not limited to, a blockchain-based distributed database, and the like. The processor referred to in the embodiments provided in the present application may be a general-purpose processor, a central processing unit, a graphics processor, a digital signal processor, a programmable logic unit, a data processing logic unit based on quantum computing, or the like, but is not limited thereto.
The technical features of the above embodiments may be arbitrarily combined, and all possible combinations of the technical features in the above embodiments are not described for brevity of description, however, as long as there is no contradiction between the combinations of the technical features, they should be considered as the scope of the description.
The foregoing examples illustrate only a few embodiments of the application and are described in detail herein without thereby limiting the scope of the application. It should be noted that it will be apparent to those skilled in the art that several variations and modifications can be made without departing from the spirit of the application, which are all within the scope of the application. Accordingly, the scope of the application should be assessed as that of the appended claims.

Claims (8)

1. A method of pareto front solution preference, the method comprising:
The terminal respectively acquires the function value of the first objective function and the function value of the second objective function; the first objective function includes a performance evaluation function, and the second objective function includes a cost evaluation function;
The terminal inputs the function value of the first objective function and the function value of the second objective function into a pre-established fuzzy inference system to obtain a steady state solution output by the fuzzy inference system;
The terminal selects a target pareto optimal solution from feasible solution sets of the first objective function and the second objective function according to the steady-state solution;
the terminal inputs the function value of the first objective function and the function value of the second objective function into a pre-established fuzzy inference system to obtain the steady-state solution output by the fuzzy inference system, and the method comprises the following steps:
The function value of the first objective function and the function value of the second objective function are input into the fuzzy reasoning system, the fuzzy reasoning system determines a gravity center real value and a membership area according to a preset membership function and a fuzzy rule base, and the steady-state solution is output according to the gravity center real value and the membership area; the gravity center real value is the gravity center real value of the membership function of the decision variable, and the membership area is the membership area corresponding to the membership function of the decision variable.
2. The method of claim 1, wherein the membership functions include a first membership function corresponding to the first objective function, a second membership function corresponding to the second objective function, and a third membership function corresponding to a decision variable; the method for determining the gravity center real value and the membership area according to the pre-established membership function and the fuzzy rule base comprises the following steps:
According to the first membership function and the second membership function, membership corresponding to the first objective function and membership corresponding to the second objective function are obtained respectively;
Determining a matching degree according to the membership degree corresponding to the first objective function and the membership degree corresponding to the second objective function;
And determining the gravity center real value and the membership area according to the matching degree, the third membership function and the fuzzy rule base.
3. The method according to any of claims 1-2, wherein the establishing of the fuzzy inference system comprises:
Determining the change rate of the true value of the ideal reasoning condition statement according to the ideal reasoning condition statement;
And establishing the fuzzy inference system according to the change rate of the former sentence of the ideal inference condition sentence and the change rate of the true value of the ideal inference condition sentence.
4. The method according to any one of claims 1-2, wherein the method further comprises:
And determining a standby objective function from the first objective function and the second objective function according to the target pareto optimal solution.
5. The method according to claim 4, wherein the method further comprises:
acquiring a plurality of pareto optimal solutions of the multidimensional target optimization problem;
and carrying out normalization processing on the pareto optimal solutions to obtain the feasible solution set.
6. A pareto front solution preference device, the device comprising:
the function value determining module is used for respectively acquiring the function value of the first objective function and the function value of the second objective function by the terminal; the first objective function includes a performance evaluation function, and the second objective function includes a cost evaluation function;
The steady-state solution determining module is used for inputting the function value of the first objective function and the function value of the second objective function into a pre-established fuzzy inference system by the terminal to obtain a steady-state solution output by the fuzzy inference system;
the optimal solution selection module is used for selecting a target pareto optimal solution from feasible solution sets of the first objective function and the second objective function according to the steady-state solution by the terminal;
The steady-state solution determining module is specifically configured to input a function value of the first objective function and a function value of the second objective function into the fuzzy inference system, determine a real gravity center value and a membership area by using the fuzzy inference system according to a membership function and a fuzzy rule base established in advance, and output the steady-state solution according to the real gravity center value and the membership area; the gravity center real value is the gravity center real value of the membership function of the decision variable, and the membership area is the membership area corresponding to the membership function of the decision variable.
7. A computer device comprising a memory and a processor, the memory storing a computer program, characterized in that the processor implements the steps of the method of any one of claims 1 to 5 when the computer program is executed.
8. A computer readable storage medium, on which a computer program is stored, characterized in that the computer program, when being executed by a processor, implements the steps of the method of any of claims 1 to 5.
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