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

Abstract

The invention discloses a multi-objective optimization method for system-level test design based on utility function. First, the influencing factors and optimization objective function are determined, then a weight vector is set, and a multi-objective optimization problem is decomposed into N sub-problems by using the weight vector. The G neighbors of the sub-problem generate new offspring individuals through crossover and mutation, use the fitness difference to select the individuals in the population with the best improvement effect for updating, and delete the dominated solution in the final generation population, that is, the influencing factors are obtained. The set of Pareto optimal solutions for vectors. By adopting the invention, the convergence effect and the uniformity of the Pareto optimal solution of the influencing factor vector can be improved while ensuring that the Pareto optimal solution set of the influencing factor vector is obtained.

Description

Multi-objective optimization method for system-level test design based on utility function

technical field

The invention belongs to the technical field of test design optimization of electronic systems, and more particularly relates to a system-level test design multi-objective optimization method based on a utility function.

Background technique

In order to reduce the difficulty of equipment maintenance in the future, the system should be designed for testability in the initial stage of design. Testability refers to the degree 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 choose a test plan, so that the fault detection rate (FDR, fault diagnosis rate), the false alarm rate (FAR, fault alarm rate) and the test costs (time, economy, etc.) Indicators that satisfy constraints at the same time or even tend to be better are constantly explored in academic and engineering fields.

In the test optimization problem, the test indicators concerned are the fault detection rate (FDR, fault diagnose rate), isolation rate, false alarm rate (FAR, fault alarm rate), test time cost (TC, time cost) and test economic cost ( PC, price cost) and so on. Increasing system testability means additional test hardware, thus affecting system weight, size, development difficulty, functional impact, and system reliability.

It is assumed that there are D total of influencing factors, which are represented by x _d , d=1,2,...,D. And the influence factor value is normalized to a variable between 0 and 1, then the influence factor vector X=[x ₁ ,...,x _D ]. Assuming that the number of objectives to be optimized is M, the objective function of each optimization objective is f _m (X), m=1,2,...,M.

The test optimization goal is to choose and set X reasonably (ie, carry out the test design reasonably, allocate resources reasonably, etc.), so that the M objective functions are minimized. In reality, it is generally impossible for M objective functions to be optimal at the same time, so this is a typical multi-objective optimization problem.

When multi-objective optimization is a minimization optimization problem, it can be expressed by the following formula, that is, it is necessary to find a suitable X to minimize all M objective functions f(X):

minimize F(X)=(f ₁ (X),f ₂ (X),…,f _M (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 composed of many Pareto (Pareto) optimal solutions, and each element in the set is called Pareto. optimal solution or non-inferior optimal solution. For the vectors F(X _i ) and F(X _j ) determined by the above formula, if the two vectors are not equal and all elements in F(X _i ) are not larger than the corresponding position elements in F(X _j ), then F(X _i ) is said to dominate F(X _j ), X _j is called a dominant solution, and X _i is called a non-dominated solution. The set consisting of all non-dominated solutions is called the Pareto optimal set.

At present, the algorithms that can solve this kind of problem include NSGA-III algorithm, particle swarm algorithm and so on. The NSGA-III algorithm is more typical and can find a more comprehensive set of non-dominated solutions. However, due to the high time complexity of the domination relationship calculation, the convergence effect is poor.

SUMMARY OF THE INVENTION

The purpose of the present invention is to overcome the deficiencies of the prior art and provide a system-level test design multi-objective optimization method based on a utility function, which can improve the convergence effect while ensuring that the Pareto optimal solution set of the influencing factor vectors is obtained. and the homogeneity of the Pareto optimal solution of the factor vector.

In order to achieve the above-mentioned object of the invention, the multi-objective optimization method of the system-level test design based on the utility function of the present invention comprises the following steps:

S1: Determine the influencing factors according to the actual situation of the electronic system, record the influencing factor vector X=[x ₁ , x ₂ ,..., x _D ], where x _d represents the normalized value of the d-th influencing factor, d=1, ₂ , . Smaller, the better the combination of influencing factors;

其中，

表示权重向量W_i的第m个元素值，i＝1,2,…,N；两两计算权重向量之间的欧式距离，对于第i个权重向量，获取与其欧式距离最小的前G个权重向量作为第i个权重向量的邻居权重向量，从而得到第i个权重向量的邻居集合B(i)＝{i₁,i₂,…,i_G}，i_g表示第i个权重向量第g个邻居权重向量的序号，g＝1,2,…,G；S2: Set N weight vectors as needed

in,

Indicates the mth element value of the weight vector Wi, _i =1,2,...,N; calculates the Euclidean distance between the weight vectors pairwise, and for the ith weight vector, obtains the first G weights with the smallest Euclidean distance from it The vector is used as the neighbor weight vector of the ith weight vector, so as to obtain the neighbor set of the ith weight vector B(i)={i ₁ ,i ₂ ,...,i _G }, i _g represents the ith weight vector g The serial number of the neighbor weight vector, g=1,2,...,G;

S3: Initialize the utility function π _i = ₁ corresponding to each weight vector Wi;

S4: Take the influencing factor vector X=[x ₁ ,...,x _D ] as the population individual, initialize the population, and denote the population size as N; let the number of iterations S=1;

S5: Calculate the values of M objective functions f _m (X _i ) corresponding to each individual in the initial population, where X _i represents the ith individual in the initial population, i=1, 2,...,N, determine the value of each objective function Ideal reference value z _m =min{f _m (X ₁ ),f _m (X ₂ ),...,f _m (X _N )}, obtain the initial ideal reference point Z=(z ₁ ,z ₂ ,...,z _M ) ^T , T represents transpose;

S6: Sort the individuals in the current population from small to large according to the utility function π _i , select the first K individuals, and denote the set of these K individuals as φ, where the size of K is determined according to actual needs;

S7: Initialize individual serial number i=1;

S8: judge whether X _i ∈ φ, if yes, go to step S9, otherwise go to step S11:

S9: Generate a random number rand in the range of [0,1], if rand<δ, and δ represents the preset selection pool probability, then let the individual selection pool E=B(i), otherwise let E={1,2 ,…,N};

S10: Use the following method to perform population evolution based on individual _Xi :

S10.1: Randomly select a serial number r from the individual selection pool E, use individuals X _i and X _r as parent individuals to perform crossover and mutation operations, and generate two offspring individuals y ₁ and y ₂ , and calculate the two offspring individuals according to the following formulas respectively. The fitness value g(y _v |W _i ,Z) of the individual generation:

Among them, v=1,2;

If g(y ₁ |W _i ,Z)≤g(y ₂ |W _i ,Z), let the evolution target individual Y=y ₁ , otherwise let the evolution target individual Y=y ₂ ;

S10.2: Calculate the values of the M objective functions f _m (Y) corresponding to the evolutionary target individual Y, and form the objective function vector F(Y)=(f ₁ (Y), f ₂ (Y),...,f _M ( Y));

S10.3: For each ideal reference value in the current ideal reference point Z=(z ₁ ,z ₂ ,...,z _M ) ^T , if z _m >f _m (Y), then let z _m =f _m (Y), otherwise do nothing;

S10.4: Calculate the fitness difference U _j between the evolution target individual Y and each individual X _j in the current population according to the following formula:

U _j =g(X _j |W _j ,Z)-g(Y|W _j ,Z)

Select the individual X _j with the largest fitness difference U _j , and use the evolution target individual Y to update it;

S11: judge whether i<N, if yes, go to step S12, otherwise go to step S13;

S12: Set the individual serial number i=i+1, and return to step S8;

S13: Determine whether the number of iterations S<S _max , S _max represents the preset maximum number of iterations, if yes, go to step S14 , otherwise go to step S17 ;

S14: Let the number of iterations S=S+1;

S15: determine whether the number of iterations is S% S _π = 0, % represents the remainder, and S _π represents the preset update period of the utility function value, if yes, go to step S16, otherwise do nothing and return to step S6;

S16: Update each utility function π _i using the following formula:

Among them, Δ _i represents the relative reduction of the fitness value of the _ith individual Xi in the current population, which is calculated by the following formula:

表示前Q代种群中第i个个体

的适应度值；Among them, g(X _i |W _i ,Z) represents the fitness value of the _ith individual Xi in the current population,

Represents the i-th individual in the previous Q generation population

The fitness value of ;

Return to step S6;

S17: Delete the dominated individual from the current population, and the remaining individual set is the Pareto optimal solution set as the influencing factor vector, and each individual corresponds to an influencing factor vector.

The present invention designs a multi-objective optimization method based on the system-level testability of the utility function. First, the influencing factors and the optimization objective function are determined, and then the weight vector is set. The weight vector is used to decompose a multi-objective optimization problem into N sub-problems, and the G of the sub-problem is used. Neighbors, generate new offspring individuals through crossover and mutation, use the fitness difference to select the individuals in the population with the best improvement effect to update, delete the dominated solution in the final generation population, and get the Pare value of the influencing factor vector. the optimal solution set. By adopting the invention, the convergence effect and the uniformity of the Pareto optimal solution of the influencing factor vector can be improved while ensuring that the Pareto optimal solution set of the influencing factor vector is obtained.

Description of drawings

Fig. 1 is the specific implementation flow chart of the multi-objective optimization method of system-level test design based on utility function of the present invention;

Fig. 2 is the flow chart of population evolution in the present invention;

3 is a schematic diagram of the objective function vector corresponding to the Pareto optimal solution obtained by adopting the present invention in the present embodiment;

Fig. 4 is the objective function vector diagram corresponding to the Pareto optimal solution that adopts NSGAIII algorithm to obtain in the present embodiment;

FIG. 5 is a comparison diagram of the convergence effect of the present invention and NSGA-III in this embodiment.

Detailed ways

The specific embodiments of the present invention are described below with reference to the accompanying drawings, so that those skilled in the art can better understand the present invention. It should be noted that, in the following description, when the detailed description of known functions and designs may dilute the main content of the present invention, these descriptions will be omitted here.

FIG. 1 is a flow chart of a specific implementation of the multi-objective optimization method for system-level test design based on the utility function of the present invention. As shown in FIG. 1 , the specific steps of the multi-objective optimization method for the system-level test design based on the utility function of the present invention include:

S101: Determine the influencing factors and optimize the objective function:

Determine the influencing factors according to the actual situation of the electronic system, denote the influencing factor vector X=[x ₁ ,...,x _D ], where x _d represents the normalized value of the d-th influencing factor, d=1,2,...,D , D represents the number of influencing factors; denote the number of objectives to be optimized as M, determine the objective function f _m (X) of each optimization objective, m=1,2,...,M, the smaller the objective function value, the greater the The better the combination.

S102: Generate weight vector:

其中，

表示权重向量W_i的第m个元素值，i＝1,2,…,N。两两计算权重向量之间的欧式距离，对于第i个权重向量，获取与其欧式距离最小的前G个权重向量作为第i个权重向量的邻居权重向量，从而得到第i个权重向量的邻居集合B(i)＝{i₁,i₂,…,i_G}，i_g表示第i个权重向量第g个邻居权重向量的序号，g＝1,2,…,G。Set N weight vectors as needed

in,

Represents the mth element value of the weight vector Wi, _i =1,2,...,N. Calculate the Euclidean distance between the weight vectors in pairs. For the ith weight vector, obtain the first G weight vectors with the smallest Euclidean distance from it as the neighbor weight vector of the ith weight vector, so as to obtain the neighbor set of the ith weight vector. _B (i)= _{ i ₁ , i ₂ , .

The formula for calculating the Euclidean distance between two weight vectors is as follows:

Among them, j=1,2,...,N.

In this embodiment, the weight vector is generated by the simplex method, and the N weight vectors W _i are evenly distributed.

The number N of weight vectors can be calculated using the following formula:

Among them, H represents a preset constant parameter.

Using N weight vectors, a multi-objective problem can be decomposed into N sub-problems, one weight vector corresponds to one sub-problem, and the multi-objective optimization problem is completed by optimizing each sub-problem (optimization in the direction of each weight vector). Optimization.

S103: Initialize the utility function:

The utility function π _i =1 corresponding to the _ith weight vector Wi is initialized.

S104: Initialize the population:

Take the influencing factor vector X=[x ₁ ,...,x _D ] as the genetic algorithm population individual, initialize the population, denote the population size as N, the _i -th individual X _i corresponds to the weight vector Wi and the utility function π _i , i=1,2,...,N. Let the number of iterations S=1.

S105: Initialize ideal reference point:

Calculate the value of M objective functions f _m (X _i ) corresponding to each individual in the initial population, X _i represents the ith individual in the initial population, i=1,2,...,N, determine the ideal reference for each objective function The value z _m =min{f _m (X ₁ ),f _m (X ₂ ),...,f _m (X _N )}, resulting in the initial ideal reference point Z=(z ₁ ,z ₂ ,...,z _M ) ^T , T stands for transpose.

S106: Screening individuals:

Sort the individuals in the current population according to the utility function π _i from small to large, select the first K individuals, and denote the set of these K individuals as φ, where the size of K is determined according to actual needs. Generally speaking, in order to take into account the efficiency and optimization effect, the value range of K is [N/4]≤K≤[N/3].

S107: Initialize the individual serial number i=1.

S108: Determine whether X _i ∈ φ, if yes, go to step S109, otherwise go to step S111.

S109: Determine the individual selection pool:

A random number rand is generated in the range of [0,1]. If rand<δ, δ represents the preset selection pool probability. In this embodiment, δ=0.9, then let the individual selection pool E=B(i), otherwise let E={1,2,...,N}, which is the entire population.

S110: Population Evolution:

Population evolution based on individual _Xi . Figure 2 is a flow chart of population evolution in the present invention. As shown in Figure 2, the specific steps of population evolution in the present invention include:

S201: Generate evolutionary target individuals:

A sequence number r is randomly selected from the individual selection pool E, and individuals X _i and X _r are used as parent individuals to perform crossover and mutation operations to generate two offspring individuals y ₁ and y ₂ , respectively, according to the following formulas to calculate the fitness of the two offspring individuals Degree value g(y _v |W _i ,Z):

where v=1,2.

If g(y ₁ |W _i ,Z)≤g(y ₂ |W _i ,Z), select the offspring individual y ₁ , that is, let the evolution target individual Y=y ₁ , otherwise let the evolution target individual Y=y ₂ .

In this embodiment, analog binary crossover is used for crossover, and polynomial mutation is used for mutation.

S202: Calculate the individual objective function vector of the evolutionary target:

Calculate the values of M objective functions f _m (Y) corresponding to the evolutionary target individual Y, and form an objective function vector F(Y)=(f ₁ (Y), f ₂ (Y),...,f _M (Y)).

S203: Update ideal reference point:

For each ideal reference value in the current ideal reference point Z=(z ₁ ,z ₂ ,...,z _M ) ^T , if z _m >f _m (Y), then let z _m =f _m (Y), Otherwise, do nothing.

S204: Update the individual:

Calculate the fitness difference U _j between the evolution target individual Y and each individual X _j in the current population according to the following formula:

U _j =g(X _j |W _j ,Z)-g(Y|W _j ,Z)

Among them, j=1,2,...,N,

Select the individual X _j with the largest fitness difference U _j , and use the evolution target individual Y to update it, that is, let X _j =Y.

S111: Determine whether i<N, if yes, go to step S112, otherwise go to step S113.

S112: Set the individual serial number i=i+1, and return to step S108.

S113: Determine whether the number of iterations S<S _max , where S _max represents the preset maximum number of iterations, if yes, go to step S114 , otherwise go to step S117 .

S114: Let the number of iterations S=S+1.

S115: Determine whether the number of iterations is S% S _π = 0, % means remainder, S _π means the preset update period of the utility function value, if yes, go to step S116, otherwise do nothing and return to step S106. Generally, the value of the update period S _π should not be too small, and can be set to 50≤S _π ≤100.

S116: Update utility function:

Each utility function π _i is updated using the following formula:

Among them, Δ _i represents the relative reduction of the fitness value of the _ith individual Xi in the current population, which is calculated by the following formula:

表示前Q代种群中第i个个体

的适应度值。Q的取值范围为20≤Q≤S_π。Among them, g(X _i |W _i ,Z) represents the fitness value of the _ith individual Xi in the current population,

Represents the i-th individual in the previous Q generation population

fitness value. The value range of Q is 20≤Q≤S _π .

Then it returns to step S106.

S117: Obtain Pareto optimal solution set:

Delete the dominated individual from the current population, and the remaining individual set is the Pareto optimal solution set as the influencing factor vector. Each individual corresponds to an influencing factor vector, and the influencing factors can be configured accordingly.

Example

In order to better illustrate the technical solution of the present invention, the following takes three-objective optimization as an example to describe the specific implementation process of the present invention. It is assumed that the optimization goal of the testability design of the electronic system is to maximize the failure detection rate FDR, expressed as f ₁ =maxmize(FDR); minimize the false alarm rate FAR, expressed as f ₂ =minimize(FAR); and the test cost C , the expression is f ₃ =minimize(C). Let f ₁ =1-maxmize(FDR), then it all turns into a minimization problem. There are many factors affecting these three goals, such as design difficulty, volume consideration, functional impact, reliability impact, etc. In this embodiment, seven influencing factors are selected, ie, the influencing factor vector X=[x ₁ , . . . , x ₇ ].

The objective function F=[f ₁ , f ₂ , f ₃ ] and the optimization problem constructed in this embodiment are as follows:

Subject to 0≤x _d ≤1,for d＝1,2,…,7

in,

According to the procedure described above, the Pareto optimal solution set is obtained, and then the objective function vector is obtained. FIG. 3 is a schematic diagram of the objective function vector corresponding to the Pareto optimal solution obtained by adopting 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 a Pareto optimal solution, but also the corresponding objective function vector can be relatively uniformly distributed on the optimal plane.

The objective function value vectors corresponding to the 91 optimal solutions are:

The influencing factor vectors for obtaining these optimal objective function vectors are:

The test designer can reasonably configure the influencing factors according to the demand importance of the three functions (detection rate, false alarm rate, fault diagnosis cost) in different occasions and according to the above operation results, so as to achieve the purpose of test optimization design.

In order to illustrate the technical effect of the present invention, the NSGAIII algorithm is used to run this example (the number of individuals is 92 and the number of generations is 500), and the results are compared with the results of the present invention. FIG. 4 is a schematic diagram of the objective function vector corresponding to the Pareto optimal solution obtained by adopting the NSGAIII algorithm in this embodiment. Comparing Figure 3 and Figure 4, it can be seen that although the NSGAIII algorithm can also be used to obtain the optimal solution, it is not completely close to the reference point, and the distribution is not uniform.

FIG. 5 is a comparison diagram of the convergence effect of the present invention and NSGA-III in this embodiment. It can be seen from Fig. 5 that the convergence of the present invention is better than that of the traditional NSGA-III algorithm.

Although the illustrative specific 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 clear that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, As long as various changes are within the spirit and scope of the present invention as defined and determined by the appended claims, these changes are obvious, and all inventions and creations utilizing the inventive concept are included in the protection list.

Claims (6)

1. a system-level test design multi-objective optimization method based on utility function, is characterized in that, comprises the following steps: S1: Determine the influencing factors according to the actual situation of the electronic system, record the influencing factor vector X=[x ₁ , x ₂ ,..., x _D ], where x _d represents the normalized value of the d-th influencing factor, d=1, ₂ , . Smaller, the better the combination of influencing factors; S2: Set N weight vectors as needed

in,

Indicates the mth element value of the weight vector Wi, _i =1,2,...,N; calculates the Euclidean distance between the weight vectors pairwise, and for the ith weight vector, obtains the first G weights with the smallest Euclidean distance from it The vector is used as the neighbor weight vector of the ith weight vector, so as to obtain the neighbor set of the ith weight vector B(i)={i ₁ ,i ₂ ,...,i _G }, i _g represents the ith weight vector g The serial number of the neighbor weight vector, g=1,2,...,G; S3: Initialize the utility function π _i = ₁ corresponding to each weight vector Wi; S4: Take the influencing factor vector X=[x ₁ ,...,x _D ] as the population individual, initialize the population, and denote the population size as N; let the number of iterations S=1; S5: Calculate the values of M objective functions f _m (X _i ) corresponding to each individual in the initial population, and determine the ideal reference value of each objective function z _m =min{f _m (X ₁ ), f _m (X ₂ ) ,...,f _m (X _N )}, obtain the initial ideal reference point Z=(z ₁ ,z ₂ ,...,z _M ) ^T , where T represents the transposition; S6: Sort the individuals in the current population from small to large according to the utility function π _i , select the first K individuals, and denote the set of these K individuals as φ, where the size of K is determined according to actual needs; S7: Initialize individual serial number i=1; S8: judge whether X _i ∈ φ, if yes, go to step S9, otherwise go to step S11: S9: Generate a random number rand in the range of [0,1], if rand<δ, and δ represents the preset selection pool probability, then let the individual selection pool E=B(i), otherwise let E={1,2 ,…,N}; S10: Use the following method to perform population evolution based on individual _Xi : S10.1: Randomly select a serial number r from the individual selection pool E, use individuals X _i and X _r as parent individuals to perform crossover and mutation operations, and generate two offspring individuals y ₁ and y ₂ , and calculate the two offspring individuals according to the following formulas respectively. The fitness value g(y _v |W _i ,Z) of the individual generation:

Among them, v=1,2; If g(y ₁ |W _i ,Z)≤g(y ₂ |W _i ,Z), let the evolution target individual Y=y ₁ , otherwise let the evolution target individual Y=y ₂ ; S10.2: Calculate the values of the M objective functions f _m (Y) corresponding to the evolutionary target individual Y, and form the objective function vector F(Y)=(f ₁ (Y), f ₂ (Y),...,f _M ( Y)); S10.3: For each ideal reference value in the current ideal reference point Z=(z ₁ ,z ₂ ,...,z _M ) ^T , if z _m >f _m (Y), then let z _m =f _m (Y), otherwise do nothing; S10.4: Calculate the fitness difference U _j between the evolution target individual Y and each individual X _j in the current population according to the following formula: U _j =g(X _j |W _j ,Z)-g(Y|W _j ,Z) Select the individual X _j with the largest fitness difference U _j , and use the evolution target individual Y to update it; S11: judge whether i<N, if yes, go to step S12, otherwise go to step S13; S12: Set the individual serial number i=i+1, and return to step S8; S13: Determine whether the number of iterations S<S _max , S _max represents the preset maximum number of iterations, if yes, go to step S14 , otherwise go to step S17 ; S14: Let the number of iterations S=S+1; S15: determine whether the number of iterations is S% S _π = 0, % represents the remainder, and S _π represents the preset update period of the utility function value, if yes, go to step S16, otherwise do nothing and return to step S6; S16: Update each utility function π _i using the following formula:

Among them, Δ _i represents the relative reduction of the fitness value of the _ith individual Xi in the current population, which is calculated by the following formula:

Among them, g(X _i |W _i ,Z) represents the fitness value of the _ith individual Xi in the current population,

Represents the i-th individual in the previous Q generation population

The fitness value of ; Return to step S6; S17: Delete the dominated individual from the current population, and the remaining individual set is the Pareto optimal solution set as the influencing factor vector, and each individual corresponds to an influencing factor vector. 2 . The multi-objective optimization method for system-level test design according to claim 1 , wherein in the step S2 , the weight vectors are generated by the simplex method, and the N weight vectors W _i are evenly distributed. 3 . 3. The system-level test design multi-objective optimization method according to claim 1, wherein in the step S2, the weight vector quantity N is calculated by the following formula:

Among them, H represents a preset constant parameter. 4 . The multi-objective optimization method for system-level test design according to claim 1 , wherein the value range of K in the step S6 is [N/4]≦K≦[N/3]. 5 . 5 . The multi-objective optimization method for system-level test design according to claim 1 , wherein the value range of the update period S _π in the step S15 is 50≦S _π ≦100. 6 . 6 . The multi-objective optimization method for system-level test design according to claim 5 , wherein the value range of Q in the step S16 is 20≦Q≦S _π . 7 .

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