CN110399968B - Multi-objective optimization method for system-level testability design based on utility function - Google Patents

Multi-objective optimization method for system-level testability design based on utility function Download PDF

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CN110399968B
CN110399968B CN201910555024.6A CN201910555024A CN110399968B CN 110399968 B CN110399968 B CN 110399968B CN 201910555024 A CN201910555024 A CN 201910555024A CN 110399968 B CN110399968 B CN 110399968B
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杨成林
姬志周
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University of Electronic Science and Technology of China
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Abstract

The invention discloses a system-level testability design multi-objective optimization method based on a utility function, which comprises the steps of determining influence factors and an optimization objective function, setting weight vectors, decomposing a multi-objective optimization problem into N subproblems by using the weight vectors, using G neighbors of the subproblems, generating new filial generation individuals through crossing and variation, selecting the individuals in a population with the best improvement effect by using a fitness difference value for updating, and deleting a dominated solution in a final population to obtain a pareto optimal solution set of the influence factor vectors. By adopting the invention, the convergence effect and the uniformity of the pareto optimal solution of the influence factor vector can be improved while ensuring that the pareto optimal solution set of the influence factor vector is obtained.

Description

Multi-objective optimization method for system-level testability design based on utility function
Technical Field
The invention belongs to the technical field of electronic system testability design optimization, and particularly relates to a system-level testability design multi-objective optimization method based on a utility function.
Background
In order to reduce the difficulty of later maintenance of the device, the system should consider testability design in the initial stage of design. Testability refers to the extent to which the state of a system can be accurately detected. In the problem of fault diagnosis for large-scale electronic equipment systems, how to select a test scheme to enable the Fault Detection Rate (FDR), the False Alarm Rate (FAR) and various overhead (time, economy and the like) indexes of testing to simultaneously meet constraint conditions tends to be better, and the method is a problem of continuous exploration in the academic and engineering fields.
In the test optimization problem, the test indexes of interest include a Fault Detection Rate (FDR), an isolation rate, a False Alarm Rate (FAR), a test Time Cost (TC), a test economic cost (PC), and the like. Increasing system testability means additional test hardware, thus affecting system weight, size, development difficulty, functional impact, and system reliability.
Assuming a total of D influencing factorsBy xdIs represented by D ═ 1,2, …, D. And normalizing the influence factor value to a variable between 0 and 1, the influence factor vector X is [ X ═ X1,…,xD]. Assuming that the number of targets to be optimized is M, the objective function of each optimization target is fm(X)，m＝1,2,…,M。
The test optimization target is to reasonably select and set X (i.e. reasonably develop testability design, reasonably allocate resources and the like) so as to minimize M target functions. In reality, it is generally impossible for M objective functions to reach the optimum simultaneously, so this is a typical multi-objective optimization problem.
When multiobjective optimization is a minimization optimization problem, it can be expressed by the following formula, i.e. it is necessary to find a suitable X to minimize all M objective functions f (X):
minimize F(X)＝(f1(X),f2(X),…,fM(X))
the essential difference from the single-objective optimization problem is that the solution of the multi-objective optimization problem is not unique, but there is a set of optimal solutions consisting of numerous Pareto (Pareto) optimal solutions, and each element in the set is called a Pareto optimal solution or a non-inferior optimal solution. For vector F (X) determined by the above formulai) And F (X)j) If the two vectors are not equal and F (X)i) All elements in the solution are not more than F (X)j) The corresponding position element in (b) is called F (X)i) Dominating F (X)j)，XjCalled the dominant solution, XiReferred to as the non-dominant solution. The set of all non-dominant solutions is called the pareto optimal set.
The current algorithms capable of solving the problems include NSGA-III type algorithm, particle swarm algorithm and the like. The NSGA-III type algorithm is typical, a relatively comprehensive non-dominated solution set can be found, and the problems of high time complexity of dominant relationship calculation, poor convergence effect and the like are solved.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a utility function-based system-level testability design multi-objective optimization method, which can improve the convergence effect and the uniformity of the pareto optimal solution of the influence factor vectors while ensuring that the pareto optimal solution set of the influence factor vectors is obtained.
In order to achieve the purpose, the utility function-based system-level testability design multi-objective optimization method comprises the following steps:
s1: determining influence factors according to the practical situation of the electronic system, and recording the influence factor vector X ═ X1,x2,…,xD]Wherein x isdA normalized value representing the D-th influencing factor, D being 1,2, …, D representing the number of influencing factors; recording the number of the targets needing to be optimized as M, and determining an objective function f of each optimized targetm(X), M is 1,2, …, M, the smaller the objective function value, the better the combination of influencing factors;
s2: setting N weight vectors as required
Figure BDA0002106638700000021
Wherein the content of the first and second substances,
Figure BDA0002106638700000022
represents a weight vector WiThe mth element value of (1), 2, …, N; calculating Euclidean distance between every two weight vectors, and for the ith weight vector, acquiring the first G weight vectors with the minimum Euclidean distance as neighbor weight vectors of the ith weight vector, thereby obtaining a neighbor set B (i) { i) } of the ith weight vector1,i2,…,iG}，igA sequence number representing the ith weight vector, wherein G is 1,2, …, and G;
s3: initializing each weight vector WiCorresponding utility function pii＝1；
S4: defining the influencing factor vector X as [ X ]1,…,xD]As a population individual, initializing the population, and recording the size of the population as N; making the iteration number S equal to 1;
s5: calculating M target functions f corresponding to each individual in the initial populationm(Xi) Value of (A), XiDenotes the ith individual in the initial population, i 1,2, …, N, and determines the ideal reference value z for each objective functionm＝min{fm(X1),fm(X2),…,fm(XN) Get the initial ideal reference point Z ═ Z (Z)1,z2,…,zM)TT represents transposition;
s6: according to utility function piiSequencing individuals in the current population from small to large, selecting the first K individuals, recording the set of the K individuals as phi, wherein the size of K is determined according to actual needs;
s7: initializing an individual serial number i to be 1;
s8: judging whether X is presentiE φ, if yes, go to step S9, otherwise go to step S11:
s9: generating a random number rand in the range of [0,1], if rand is less than delta, delta represents a preset selection pool probability, making the individual selection pool E ═ B (i), and otherwise, making E ═ 1,2, …, N };
s10: based on individuals X using the following methodiCarrying out population evolution:
s10.1: randomly selecting a serial number r from an individual selection pool E, and enabling an individual XiAnd XrPerforming crossover and mutation operation as parent individuals to generate two child individuals y1And y2Calculating the fitness values g (y) of the two filial generation individuals according to the following formula respectivelyv|Wi,Z)：
Figure BDA0002106638700000031
Wherein v is 1, 2;
if g (y)1|Wi,Z)≤g(y2|WiZ), let Y be Y for the evolution target individual1Otherwise, let the evolution target individual Y be Y2；
S10.2: calculating M target functions f corresponding to evolution target individuals YmThe value of (Y) constitutes an objective function vector f (Y) ═ f1(Y),f2(Y),…,fM(Y))；
S10.3: for the current ideal reference point Z ═ Z (Z)1,z2,…,zM)TIf z is each ideal reference value ofm＞fm(Y) then let zm＝fm(Y), otherwise do nothing;
s10.4: calculating an evolution target individual Y and each individual X in the current population according to the following formulajIs not suitable forj：
Uj＝g(Xj|Wj,Z)-g(Y|Wj,Z)
Selecting a fitness difference value UjMaximum individual XjUpdating the evolution target individual Y by adopting the evolution target individual Y;
s11: judging whether i is less than N, if so, entering step S12, otherwise, entering step S13;
s12: returning to step S8 when the individual number i is i + 1;
s13: judging whether the iteration number S is less than Smax，SmaxRepresenting a preset maximum iteration number, if so, entering step S14, otherwise, entering step S17;
s14: making the iteration number S equal to S + 1;
s15: judging whether the iteration times is S percent S π0,% represents remainder, SπIf yes, the step S16 is carried out, otherwise, no operation is carried out, and the step S6 is returned;
s16: for each utility function, pi, the following formula is usediUpdating:
Figure BDA0002106638700000041
wherein, DeltaiRepresents the ith individual X in the current populationiThe relative reduction number of the fitness value is calculated by adopting the following formula:
Figure BDA0002106638700000042
wherein g (X)i|WiZ) denotes the ith individual X in the current populationiThe value of the fitness value of (a) is,
Figure BDA0002106638700000043
represents the ith individual in the prior Q generation population
Figure BDA0002106638700000044
A fitness value of;
returning to step S6;
s17: and deleting the dominated individual from the current population, wherein the rest individual set is a pareto optimal solution set serving as the influence factor vector, and each individual corresponds to one influence factor vector.
The invention relates to a system-level testability design multi-objective optimization method based on a utility function, which comprises the steps of firstly determining influence factors and an optimization objective function, then setting weight vectors, decomposing a multi-objective optimization problem into N subproblems by using the weight vectors, using G neighbors of the subproblems, generating new filial individuals through crossing and variation, selecting individuals in a population with the best improvement effect by using a fitness difference value for updating, and deleting a dominated solution in a final population to obtain a pareto optimal solution set of the influence factor vectors. By adopting the invention, the convergence effect and the uniformity of the pareto optimal solution of the influence factor vector can be improved while ensuring that the pareto optimal solution set of the influence factor vector is obtained.
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FIG. 1 is a flow chart of an embodiment of the utility function based system level testability design multi-objective optimization method of the present invention;
FIG. 2 is a flow chart of population evolution in the present invention;
fig. 3 is a schematic diagram of a target function vector corresponding to the Pareto optimal solution obtained by the present invention in this embodiment;
fig. 4 is a schematic diagram of a target function vector corresponding to the Pareto optimal solution obtained by using the NSGAIII algorithm in this embodiment;
FIG. 5 is a graph comparing the convergence effect of the present invention and NSGA-III in this example.
Detailed Description
The following description of the embodiments of the present invention is provided in order to better understand the present invention for those skilled in the art with reference to the accompanying drawings. It is to be expressly noted that in the following description, a detailed description of known functions and designs will be omitted when it may obscure the subject matter of the present invention.
FIG. 1 is a flowchart of an embodiment of the utility function-based system-level testability design multi-objective optimization method of the present invention. As shown in FIG. 1, the utility function-based system-level testability design multi-objective optimization method of the invention specifically comprises the following steps:
s101: determining influencing factors and optimizing an objective function:
determining influence factors according to the practical situation of the electronic system, and recording the influence factor vector X ═ X1,…,xD]Wherein x isdA normalized value representing the D-th influencing factor, D being 1,2, …, D representing the number of influencing factors; recording the number of the targets needing to be optimized as M, and determining an objective function f of each optimized targetm(X), M is 1,2, …, M, and the smaller the objective function value, the better the combination of influencing factors.
S102: generating a weight vector:
setting N weight vectors as required
Figure BDA0002106638700000051
Wherein the content of the first and second substances,
Figure BDA0002106638700000052
represents a weight vector WiI-1, 2, …, N. Calculating Euclidean distance between every two weight vectors, and for the ith weight vector, acquiring the first G weight vectors with the minimum Euclidean distance as neighbor weight vectors of the ith weight vector, thereby obtaining a neighbor set B (i) { i) } of the ith weight vector1,i2,…,iG}，igThe index indicating the ith weight vector is the G th neighbor weight vector, G being 1,2, …, G.
The calculation formula of the euclidean distance between the two weight vectors is as follows:
Figure BDA0002106638700000053
where j is 1,2, …, N.
In this embodiment, the weight vectors are generated by simplex method, and N weight vectors WiAre uniformly distributed.
The number of weight vectors N can be calculated using the following formula:
Figure BDA0002106638700000054
wherein H represents a preset constant parameter.
A multi-objective problem can be decomposed into N subproblems by adopting N weight vectors, one weight vector corresponds to one subproblem, and optimization of the multi-objective optimization problem is further completed by optimizing each subproblem (optimization in the direction of each weight vector).
S103: initializing utility function:
initializing the ith weight vector WiCorresponding utility function pii＝1。
S104: initializing a population:
defining the influencing factor vector X as [ X ]1,…,xD]As a genetic algorithm population individual, initializing the population, recording the size of the population as N, and recording the ith individual XiAnd a weight vector WiSum utility function piiCorrespondingly, i is 1,2, …, N. Let the iteration number S be 1.
S105: initializing the ideal reference point:
calculating M target functions f corresponding to each individual in the initial populationm(Xi) Value of (A), XiDenotes the ith individual in the initial population, i 1,2, …, N, and determines the ideal reference value z for each objective functionm＝min{fm(X1),fm(X2),…,fm(XN) Get the initial ideal reference point Z ═ Z (Z)1,z2,…,zM)TAnd T denotes transposition.
S106: screening individuals:
according to utility function piiAnd (4) sequencing the individuals in the current population from small to large, selecting the first K individuals, and recording the set of the K individuals as phi, wherein the size of K is determined according to actual needs. In general, K is in the range of [ N/4] for efficiency and optimization]≤K≤[N/3]。
S107: the initialization individual number i is 1.
S108: judging whether X is presentiE phi, if yes, go to step S109, otherwise go to step S111.
S109: determining an individual selection pool:
and generating a random number rand in the range of [0,1], if rand is less than delta, and delta represents a preset selection pool probability, wherein in the embodiment, delta is 0.9, making the individual selection pool E be B (i), otherwise, making E be {1,2, …, N }, namely the whole population.
S110: population evolution:
based on individual XiAnd (5) carrying out population evolution. FIG. 2 is a flow chart of population evolution in the present invention. As shown in fig. 2, the specific steps of population evolution in the present invention include:
s201: generating an evolution target individual:
randomly selecting a serial number r from an individual selection pool E, and enabling an individual XiAnd XrPerforming crossover and mutation operation as parent individuals to generate two child individuals y1And y2Calculating the fitness values g (y) of the two filial generation individuals according to the following formula respectivelyv|Wi,Z)：
Figure BDA0002106638700000061
Wherein v is 1, 2.
If g (y)1|Wi,Z)≤g(y2|WiAnd Z), then selecting the offspring individuals y1That is, let the evolution target individual Y be Y1Otherwise, let the evolution target individual Y be Y2。
In this embodiment, the crossover uses analog binary crossover, and the variation uses polynomial variation.
S202: calculating an evolution target individual objective function vector:
calculating M target functions f corresponding to evolution target individuals YmThe value of (Y) constitutes an objective function vector f (Y) ═ f1(Y),f2(Y),…,fM(Y))。
S203: updating the ideal reference point:
for the current ideal reference point Z ═ Z (Z)1,z2,…,zM)TIf z is each ideal reference value ofm＞fm(Y) then let zm＝fm(Y), otherwise, no operation is performed.
S204: updating individuals:
calculating an evolution target individual Y and each individual X in the current population according to the following formulajIs not suitable forj：
Uj＝g(Xj|Wj,Z)-g(Y|Wj,Z)
Where j is 1,2, …, N,
Figure BDA0002106638700000071
Figure BDA0002106638700000072
selecting a fitness difference value UjMaximum individual XjUpdating it by using evolution target individual Y, i.e. ordering Xj＝Y。
S111: and judging whether i is less than N, if so, entering step S112, and otherwise, entering step S113.
S112: the individual number i is set to i +1, and the process returns to step S108.
S113: judging whether the iteration number S is less than Smax，SmaxRepresenting a preset maximum number of iterations, and if so, proceeding to step S114, otherwise, proceeding to step S117。
S114: let S be S + 1.
S115: judging whether the iteration times is S percent S π0,% represents remainder, SπAnd (4) if the updating period of the preset utility function value is represented, the step S116 is carried out, otherwise, no operation is carried out, and the step S106 is returned. Update period SπThe value of (A) is generally not too small and can be set to 50. ltoreq. Sπ≤100。
S116: updating the utility function:
for each utility function, pi, the following formula is usediUpdating:
Figure BDA0002106638700000081
wherein, DeltaiRepresents the ith individual X in the current populationiThe relative reduction number of the fitness value is calculated by adopting the following formula:
Figure BDA0002106638700000082
wherein g (X)i|WiZ) denotes the ith individual X in the current populationiThe value of the fitness value of (a) is,
Figure BDA0002106638700000083
represents the ith individual in the prior Q generation population
Figure BDA0002106638700000084
The fitness value of (a). The value range of Q is more than or equal to 20 and less than or equal to Sπ。
And then returns to step S106.
S117: obtaining a Pareto optimal solution set:
and deleting the dominated individual from the current population, wherein the rest individual set is a pareto optimal solution set serving as the influence factor vector, and each individual corresponds to one influence factor vector, so that the influence factors can be configured accordingly.
Examples
In order to better explain the technical scheme of the invention, the following takes three-target optimization as an example to explain the concrete implementation process of the invention. Assuming that the optimization target of the electronic system testability design is to maximize the fault detection rate FDR, the expression is f1Maxmize (fdr); minimizing false alarm rate FAR, the expression is f2Minimize (far); and a test cost C, expressed as f3Minimize (c). Let f11-maxmize (fdr), all translate to minimization problems. There are many factors that affect the three targets, such as design difficulty, volume consideration, function influence, reliability influence, and the like, and in this embodiment, 7 influencing factors are selected, that is, the influencing factor vector X is [ X ═ in1,…,x7]。
The objective function F ═ F constructed in this embodiment1,f2,f3]And the optimization problem is as follows:
Figure BDA0002106638700000085
Figure BDA0002106638700000086
Figure BDA0002106638700000087
Subject to 0≤xd≤1,for d＝1,2,…,7
wherein the content of the first and second substances,
Figure BDA0002106638700000091
and (4) obtaining a pareto optimal solution set according to the flow implementation, and then obtaining an objective function vector. Fig. 3 is a schematic diagram of objective function vectors corresponding to Pareto optimal solutions obtained by the present invention in this embodiment. As shown in fig. 3, according to the method of the present invention, the found solution is not only the pareto optimal solution, but also the corresponding objective function vectors can be distributed more uniformly on the optimal plane.
The objective function value vectors corresponding to the 91 optimal solutions are respectively:
Figure BDA0002106638700000092
Figure BDA0002106638700000101
Figure BDA0002106638700000111
Figure BDA0002106638700000121
the influence factor vectors of the optimal objective function vectors are obtained as follows:
Figure BDA0002106638700000122
Figure BDA0002106638700000131
Figure BDA0002106638700000141
Figure BDA0002106638700000151
the testability designer can reasonably configure influence factors according to the requirement importance of the three functions (detection rate, false alarm rate and fault diagnosis cost) in different occasions and according to the operation results, and the purpose of testability optimization design is achieved.
To illustrate the technical effect of the present invention, this example (number of individuals 92, generation number 500) was run using NSGAIII algorithm, and the results were compared with the results of the present invention. Fig. 4 is a schematic diagram of an objective function vector corresponding to the Pareto optimal solution obtained by using the NSGAIII algorithm in this embodiment. Comparing fig. 3 and fig. 4, it can be seen that although the NSGAIII algorithm can also be used to obtain the optimal solution, the optimal solution is not completely close to the reference point, and the distribution is not uniform.
FIG. 5 is a graph comparing the convergence effect of the present invention and NSGA-III in this example. It can be seen from fig. 5 that the convergence of the present invention is better than the conventional NSGA-type III algorithm.
Although illustrative embodiments of the present invention have been described above to facilitate the understanding of the present invention by those skilled in the art, it should be understood that the present invention is not limited to the scope of the embodiments, and various changes may be made apparent to those skilled in the art as long as they are within the spirit and scope of the present invention as defined and defined by the appended claims, and all matters of the invention which utilize the inventive concepts are protected.

Claims (6)

1. A system-level testability design multi-objective optimization method based on a utility function is characterized by comprising the following steps:
s1: determining influence factors according to the practical situation of the electronic system, and recording the influence factor vector X ═ X1,x2,…,xD]Wherein x isdA normalized value representing the D-th influencing factor, D being 1,2, …, D representing the number of influencing factors; recording the number of the targets needing to be optimized as M, and determining an objective function f of each optimized targetm(X), M is 1,2, …, M, the smaller the objective function value, the better the combination of influencing factors;
s2: setting N weight vectors as required
Figure FDA0003259965460000011
Wherein the content of the first and second substances,
Figure FDA0003259965460000012
represents a weight vector WiThe mth element value of (1), 2, …, N; calculating Euclidean distance between every two weight vectors, and for the ith weight vector, acquiring the first G weight vectors with the minimum Euclidean distance as neighbor weight vectors of the ith weight vector, thereby obtaining a neighbor set B (i) { i) } of the ith weight vector1,i2,…,iG}，igA sequence number representing the ith weight vector, wherein G is 1,2, …, and G;
s3: initializing each weight vector WiCorresponding utility function pii＝1；
S4: defining the influencing factor vector X as [ X ]1,…,xD]As a population individual, initializing the population, and recording the size of the population as N; making the iteration number S equal to 1;
s5: calculating M target functions f corresponding to each individual in the initial populationm(Xi) Determining an ideal reference value z for each objective functionm＝min{fm(X1),fm(X2),…,fm(XN) Get the initial ideal reference point Z ═ Z (Z)1,z2,…,zM)TT represents transposition;
s6: according to utility function piiSequencing individuals in the current population from small to large, selecting the first K individuals, recording the set of the K individuals as phi, wherein the size of K is determined according to actual needs;
s7: initializing an individual serial number i to be 1;
s8: judging whether X is presentiE φ, if yes, go to step S9, otherwise go to step S11:
s9: generating a random number rand in the range of [0,1], if rand is less than delta, delta represents a preset selection pool probability, making the individual selection pool E ═ B (i), and otherwise, making E ═ 1,2, …, N };
s10: based on individuals X using the following methodiCarrying out population evolution:
s10.1: randomly selecting a serial number r from an individual selection pool E, and enabling an individual XiAnd XrAs a parent individualLine crossing and mutation operations, resulting in two offspring individuals y1And y2Calculating the fitness values g (y) of the two filial generation individuals according to the following formula respectivelyv|Wi,Z)：
Figure FDA0003259965460000013
Wherein v is 1, 2;
if g (y)1|Wi,Z)≤g(y2|WiZ), let Y be Y for the evolution target individual1Otherwise, let the evolution target individual Y be Y2；
S10.2: calculating M target functions f corresponding to evolution target individuals YmThe value of (Y) constitutes an objective function vector f (Y) ═ f1(Y),f2(Y),…,fM(Y))；
S10.3: for the current ideal reference point Z ═ Z (Z)1,z2,…,zM)TIf z is each ideal reference value ofm＞fm(Y) then let zm＝fm(Y), otherwise do nothing;
s10.4: calculating an evolution target individual Y and each individual X in the current population according to the following formulajIs not suitable forj：
Uj＝g(Xj|Wj,Z)-g(Y|Wj,Z)
Selecting a fitness difference value UjMaximum individual XjUpdating the evolution target individual Y by adopting the evolution target individual Y;
s11: judging whether i is less than N, if so, entering step S12, otherwise, entering step S13;
s12: returning to step S8 when the individual number i is i + 1;
s13: judging whether the iteration number S is less than Smax，SmaxRepresenting a preset maximum iteration number, if so, entering step S14, otherwise, entering step S17;
s14: making the iteration number S equal to S + 1;
s15: judging whether the number of iterations isS％Sπ0,% represents remainder, SπIf yes, the step S16 is carried out, otherwise, no operation is carried out, and the step S6 is returned;
s16: for each utility function, pi, the following formula is usediUpdating:
Figure FDA0003259965460000021
wherein, DeltaiRepresents the ith individual X in the current populationiThe relative reduction number of the fitness value is calculated by adopting the following formula:
Figure FDA0003259965460000022
wherein g (X)i|WiZ) denotes the ith individual X in the current populationiThe value of the fitness value of (a) is,
Figure FDA0003259965460000023
represents the ith individual in the prior Q generation population
Figure FDA0003259965460000024
A fitness value of;
returning to step S6;
s17: and deleting the dominated individual from the current population, wherein the rest individual set is a pareto optimal solution set serving as the influence factor vector, and each individual corresponds to one influence factor vector.
2. The system-level design-for-test multi-objective optimization method of claim 1, wherein the weight vectors in step S2 are generated by simplex method, and N weight vectors W are generatediAre uniformly distributed.
3. The system-level design multi-objective optimization method of claim 1, wherein the weight vector quantity N in step S2 is calculated by using the following formula:
Figure FDA0003259965460000031
wherein H represents a preset constant parameter.
4. The system-level testability design multi-objective optimization method of claim 1, wherein the value range of K in step S6 is [ N/4] ≦ K ≦ N/3 ].
5. The system-level testability design multi-objective optimization method of claim 1, wherein the update period S in step S15 is set as the period SπHas a value range of 50 to Sπ≤100。
6. The system-level testability design multi-objective optimization method of claim 5, wherein the value range of Q in step S16 is 20-Q-Sπ。
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