Weapon target allocation optimization method based on improved biogeography

Abstract

The present invention proposes a weapon target allocation optimization method based on improved biogeography, which includes the following steps: constructing a WTA optimization model based on maximizing the cost-effectiveness ratio; setting the population size popsize, randomly generating the initial solution of the population; encoding the population using an integer-based matrix coding strategy; calculating the fitness function value, sorting from large to small, and retaining the solutions corresponding to the first k larger fitness values; designing a migration operator based on a cosine dynamic adaptive strategy to perform a migration operation; designing a mutation operator based on a symbiotic strategy to perform a mutation operation; recalculating the fitness function value Z, sorting from large to small, and saving the current optimal solution; judging whether the maximum number of iterations has been reached, and outputting the result if it has; otherwise returning to the fifth step. The present invention can better coordinate the local search and global search capabilities of the optimization algorithm, improve the accuracy and efficiency of solving the WTA problem, and provide method support for the development of command and control systems.

Description

Technical Field

[0001] The present invention relates to the field of swarm intelligence optimization technology, and in particular to a weapon target allocation optimization method based on improved biogeography. Background Art

[0002] Weapon target assignment (WTA) is a core problem in the field of command and control. It is also a typical NP-hard problem with the characteristic of "curse of dimensionality". The size of the solution space of the problem grows exponentially with the increase in the number of weapon units and the number of incoming targets.

[0003] For solving WTA problems, heuristic algorithms with good adaptability to NP-hard problems are currently used, such as genetic algorithms, particle swarm algorithms, ant colony algorithms, bee colony algorithms, etc. However, when solving large-scale WTA problems, existing algorithms have a certain degree of high algorithm complexity, slow model solution speed, and low solution accuracy. The biogeography-based optimization (BBO) algorithm was proposed by Simon "Biogeography-based optimization" (IEEE transactions on evolutionary computation, 2008, 12 (6): 702-713.). This algorithm has attracted the attention of many scholars because of its advantage of being able to enhance information exchange and sharing through immigration rate and emigration rate.

[0004] In terms of the optimization of the BBO algorithm, it is usually divided into three categories: one is optimization based on migration and mutation operations, the second is hybridization of other intelligent algorithms or precise algorithms, and the third is the establishment of multiple populations or local topologies. Ma Ping et al.'s "Biogeography Optimization Algorithm Based on Neighborhood Gravitational Learning" (Computer Engineering and Applications, 2018, 54(22): 35-41.) proposed a biogeography optimization algorithm based on neighborhood gravity learning, which makes full use of the habitat information provided by the neighborhood and uses gravity learning to improve the habitat of immigrants. At the same time, an adaptive Gaussian mutation mechanism is proposed to improve the convergence of the algorithm and enhance the search ability. Zhang Xinming et al.'s "Biogeography Optimization Algorithm of Differential Migration and Optimal Mutation" (Small Microcomputer Systems, 2018, 39(06): 1168-1177.) proposed a biogeography optimization algorithm of differential migration and optimal mutation, which has strong stability and fast convergence speed. Luo Ruihan et al.'s "Firepower Allocation Scheme Optimization Based on Improved BBO Algorithm" (Journal of Nanjing University of Aeronautics and Astronautics, 2020, 52(06): 897-902.) adds a three-dimensional mutation operation to the BBO algorithm, optimizes the firepower allocation scheme, and increases the global search capability for the scheme. Summary of the invention

[0005] The present invention provides a weapon target allocation optimization method based on improved biogeography, which specifically comprises the following steps:

[0006] Step 1: Construct a WTA optimization model based on maximizing the cost-effectiveness ratio

[0007] Suppose there are n weapon platforms, i = 1, 2, ..., n, i is the ordinal number of the weapon platform, and m incoming targets are intercepted, j = 1, 2, ..., m, j is the ordinal number of the incoming target, and the number of weapons equipped on each weapon platform is C i ; V j is the jth target value; Q ij R is the shooting advantage of the i-th weapon platform against the j-th incoming target; ij is the attack risk of the i-th weapon platform on the j-th incoming target; ij The number of weapons assigned to the jth incoming target by the i-th weapon platform is used to establish the multi-objective function as follows:

[0008]

[0009] In the formula, Z 1 represents the shooting benefit function, Z 2 represents the combat cost function, λ is the weight parameter given by the expert;

[0010] In order to transform multi-objective optimization into common single-objective optimization, a WTA optimization model based on maximizing the cost-effectiveness ratio is constructed, as shown in formula (2):

[0011]

[0012] Where Z is the fitness function value;

[0013] The constraints in the model have the following meanings:

[0014] ① Each weapon platform can attack multiple incoming targets at the same time, and multiple weapon platforms can attack the same incoming target, with at least one weapon assigned to each incoming target;

[0015] ② Each weapon platform is equipped with multiple weapons. The weapons assigned by each weapon platform to the incoming target at the same time cannot exceed the total number of weapons of the weapon platform;

[0016] Step 2: Set the population size popsize and randomly generate the initial solution of the population.

[0017] The initial solution is the first randomly generated x ij ;

[0018] Step 3: Encode the population using an integer-based matrix encoding strategy

[0019] Let X ij Represents the allocation scheme and uses formula (3) to encode the population:

[0020]

[0021] Step 4 Calculate the fitness function value Z according to formula (2), sort them from large to small, and retain the solutions x corresponding to the first k larger fitness values ij ;

[0022] Step 5: Design a migration operator based on the cosine dynamic adaptive strategy to perform migration operations

[0023] Step 6: Design a mutation operator based on the symbiosis strategy and perform mutation operations

[0024] In order to improve the quality of the solution, the mutualistic symbiosis idea of ​​the symbiotic biological search algorithm is introduced: for a solution in the ecosystem, another solution in the ecosystem is randomly selected as the symbiotic object, and the favorable factors of each other are absorbed for co-evolution; since the number of solution components in different rows and columns in the matrix encoding of formula (3) is different, and they constrain each other, in order to reflect the co-evolution, the idea of ​​interaction in the symbiotic biological search algorithm is introduced, and a mutation operator based on the symbiotic strategy is proposed, as shown in formula (8):

[0025]

[0026] Where X best represents the optimal solution, X i represents the i-th solution selected, X i(new) represents the new solution generated, BF is the profit numerator, MV represents the relationship between solutions i and j, and the value of MV can be calculated by formula (8); n is the number of elements in the i-th row of the matrix of formula (3);

[0027] Step 7 Recalculate the fitness function value Z, sort from large to small, and save the current optimal solution

[0028] Step 8: Determine whether the maximum number of iterations G has been reached max , output the result; otherwise return to Step 5.

[0029] In one embodiment of the present invention, Step 5 specifically includes the following steps:

[0030] Step 5.1 Dynamic selection of habitat migration operators

[0031] Habitat dynamic selection migration operator:

[0032]

[0033]

[0034] Among them, P k is the probability that the kth solution is selected for migration; popsize is the population size; μ k is the emigration rate of the kth habitat, P k , μ k The a in the upper right corner represents the power of a; μ l is the emigration rate of any l-th habitat, l represents the l-th solution, l = 1, 2, ..., popsize; let a be the selection pressure factor, pd max is the maximum modification probability, pd min is the minimum modification probability; G is the current iteration number, G max is the maximum number of iterations;

[0035] Step 5.2 Differential mutation migration operator

[0036] During the Gth iteration, G does not exceed the maximum number of iterations G max , G max Determined according to the specific situation, by randomly selecting SIV features r1, r2, r3 of three different habitats, and satisfying r1≠r2≠r3, a new mutation vector generated by the three features is obtained:

[0037] X ij =X r1j +F*(X r2j -X r3j ) (6)

[0038] Where X ij is the solution of the jth dimension of the ith population; X r1j , X r2j , X r3j are the solutions of the jth dimension of the three different populations selected respectively; F is the scaling factor;

[0039] Step 5.3 Cosine dynamic adaptive selection strategy

[0040] The cosine dynamic adaptive selection strategy is designed as follows:

[0041]

[0042] Where β is the cosine dynamic adaptive selection factor;

[0043] When the random probability is greater than β, the differential mutation migration operator is used for modification; when the random probability is less than β, the dynamic selection migration operator is used to modify the out-migration location;

[0044] In the early stage of iteration, the probability that the random probability is less than β is relatively high, and the dynamic selection migration operator is used for modification; in the later stage, the probability that the random probability is greater than β is relatively high, and the differential mutation migration operator is used for modification.

[0045] Since WTA has high requirements for timeliness and accuracy, dynamic selection and cosine adaptive differential evolution operators are used to improve the migration operation, and the idea of ​​reverse learning is used to optimize the mutation operation. An improved biogeography-based optimization (IBBO) algorithm is proposed. Simulation examples show that the algorithm better coordinates the capabilities of local search and global search, and can effectively solve large-scale WTA problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 The cosine dynamic adaptive factor variation trend diagram is shown;

[0047] Figure 2 Shows the fitness curves of different algorithms. DETAILED DESCRIPTION

[0048] In order to make the objectives, technical routes and advantages of the present invention more clear, the present invention is further described in detail below with reference to the accompanying drawings and embodiments.

[0049] Step 1: Construct a WTA optimization model based on maximizing the cost-effectiveness ratio

[0050] In order to improve the effect of weapon use, it is necessary to comprehensively consider the target value, shooting advantage and shooting risk, and to maximize the shooting benefit and minimize the combat cost as the goal, and to build a WTA optimization model based on maximizing the cost-effectiveness ratio. Suppose there are n weapon platforms (i = 1, 2, ..., n, i is the ordinal number of the weapon platform), intercepting m (j = 1, 2, ..., m, j is the ordinal number of the incoming target) incoming targets, and each weapon platform is equipped with a weapon number C i ; V j is the jth target value; Q ij R is the shooting advantage of the i-th weapon platform against the j-th incoming target; ij is the attack risk of the i-th weapon platform on the j-th incoming target; ij The number of weapons assigned to the jth incoming target by the i-th weapon platform is used to establish the multi-objective function as follows:

[0051]

[0052] In the formula, Z 1 represents the shooting benefit function, Z 2represents the combat cost function, and λ is the weight parameter given by the expert.

[0053] In order to transform multi-objective optimization into common single-objective optimization, a WTA optimization model based on maximizing the cost-effectiveness ratio is constructed, as shown in formula (2):

[0054]

[0055] Where Z is the fitness function value.

[0056] The constraints in the model have the following meanings:

[0057] ① Each weapon platform can attack multiple incoming targets at the same time, and multiple weapon platforms can attack the same incoming target. Each incoming target is assigned at least one weapon.

[0058] ② Each weapon platform is equipped with multiple weapons. The weapons allocated by each weapon platform to the incoming target at the same time cannot exceed the total number of weapons of the weapon platform.

[0059] Step 2: Set the population size popsize and randomly generate the initial solution of the population.

[0060] The initial solution is the first randomly generated x ij .

[0061] Step 3: Encode the population using an integer-based matrix encoding strategy

[0062] Let X ij Represents the allocation scheme and uses formula (3) to encode the population:

[0063]

[0064] Step 4 Calculate the fitness function value Z according to formula (2), sort them from large to small, and retain the solutions x corresponding to the first k larger fitness values ij .

[0065] Step 5: Design a migration operator based on the cosine dynamic adaptive strategy to perform migration operations

[0066] The details are as follows.

[0067] Step 5.1 Dynamic selection of habitat migration operators

[0068] The inspiration of the biogeographic optimization algorithm comes from the species migration model in biogeography. In nature, species often migrate from one island to another, and in this process, new species are also generated and existing species become extinct. Islands are called "habitats" in the BBO algorithm. Habitats that are considered suitable for species to live have a higher habitat suitability index (HSI). Factors related to this index include rainfall, vegetation diversity, terrain feature diversity, temperature, etc. These factors are called suitability index variables (SIV). SIV can be regarded as the independent variable of the habitat, and HSI can be regarded as the dependent variable. Habitats with high HSI have a large number of species, low immigration rate and high out-migration rate; habitats with low HSI have a small number of species, high immigration rate and low out-migration rate. At the same time, the immigration of new species may also increase the HSI of the habitat, because the suitability of the habitat is proportional to its biodiversity.

[0069] In the migration mechanism of the BBO algorithm, the probability of a habitat being selected for migration operation is proportional to the immigration rate, while the source of its migration is proportional to the out-migration rate. In the selection, the original algorithm mainly adopts the roulette method for selection. This method can retain the better habitat with a high probability, and its SIV characteristics can be selected and the poorer individuals can be migrated, so that the solution is continuously updated and optimized. However, it is also easy to have a low diversity of the population, which will cause the local convergence of the algorithm. Therefore, in different stages, different selection pressures need to be specified when performing migration operations. Reduce the selection pressure in the early stage so that the habitat with a lower HIS value can participate in the subsequent iterative optimization and maintain the diversity of the population. Increase the selection pressure appropriately in the later stage so that the solution can converge faster and get closer to the optimal solution. The present invention improves the migration operation of the original algorithm and proposes a habitat dynamic selection migration operator:

[0070]

[0071]

[0072] Among them, P k is the probability that the kth solution is selected for migration; popsize is the population size; μ k is the emigration rate of the kth habitat, P k , μ k The a in the upper right corner represents the power of a; μ l is the emigration rate of any l-th habitat, l represents the l-th solution (l = 1, 2, ..., popsize). Let a be the selection pressure factor, pd max is the maximum modification probability, pdmin is the minimum modification probability; G is the current iteration number, G max is the maximum number of iterations.

[0073] Step 5.2 Differential mutation migration operator

[0074] The differential evolution algorithm simulates hybridization and mutation in genetics to design genetic operators. It is similar to the genetic algorithm in structure, but slightly different in mutation operation. Differential evolution has a strong exploration ability, which can make up for the singleness of the BBO algorithm in the migration process and enhance the exploration ability of the BBO algorithm. Therefore, the present invention introduces the mutation idea in the differential evolution algorithm into the migration process of the BBO algorithm and proposes a differential mutation migration operator.

[0075] During the Gth iteration, G does not exceed the maximum number of iterations G max , G max According to the specific situation, by randomly selecting SIV features r1, r2, r3 of three different habitats, and satisfying r1≠r2≠r3, a new mutation vector generated by the three features can be obtained:

[0076] X ij =X r1j +F*(X r2j -X r3j ) (6)

[0077] Where X ij is the solution of the jth dimension of the ith population; X r1j , X r2j , X r3j are the solutions of the j-th dimension of three different populations selected respectively; F is the scaling factor.

[0078] Step 5.3 Cosine dynamic adaptive selection strategy

[0079] In order to further improve the search capability of the BBO algorithm at different stages, the dynamic selection migration operator is mainly used in the early stage, and the differential mutation migration operator is mainly used in the later stage. The present invention designs a cosine dynamic adaptive selection strategy, which presents a dynamic oscillating cosine change trend and can better integrate the dynamic selection migration operator and the differential mutation migration operator. The cosine dynamic adaptive selection strategy is as follows:

[0080]

[0081] Where β is the cosine dynamic adaptive selection factor.

[0082] Figure 1It reflects the changing trend of the cosine dynamic adaptive factor, presenting the dynamic change law of oscillation and the overall downward convergence trend. When the random probability is greater than β, the differential mutation migration operator is used for modification; when the random probability is less than β, the dynamic selection migration operator is used to modify the migration site.

[0083] In the early stage of iteration, the probability of random probability being less than β is relatively high, and the dynamic selection migration operator is mainly used for modification. This operator is conducive to maintaining the diversity of the population and avoiding falling into the local optimum and failing to find the global optimal solution; in the later stage, the probability of random probability being greater than β is relatively high, and the differential mutation migration operator is mainly used for modification. This operator is more conducive to finding the global optimal solution and is conducive to the convergence of the algorithm. The overall oscillation trend of the cosine dynamic adaptive strategy can avoid the singleness of the modification strategy, and is also conducive to the exploration of unknown habitats, so that individuals in the habitat that are not particularly excellent have a certain probability of survival, avoiding the phenomenon of similarity and repetition caused by all being modified and replaced.

[0084] Step 6: Design a mutation operator based on the symbiosis strategy and perform mutation operations

[0085] It can be found that the mutation operator in the original BBO algorithm mainly mutates the poor solutions after HSI sorting (the solutions include the solutions corresponding to the first k larger fitness values, intermediate solutions and poor solutions) according to the mutation probability, but this mutation is random, which is conducive to improving the diversity of solutions, but the possibility of providing excellent solutions is small. The mutualistic symbiosis idea of ​​the symbiotic biological search algorithm means that for a certain solution in the ecosystem, another solution in the ecosystem is randomly selected as the symbiotic object, absorbing the favorable factors between each other for co-evolution and mutual benefit. Since the number of solution components in different rows and columns in the matrix encoding of formula (3) is different, and they constrain each other, in order to reflect co-evolution, the present invention introduces the idea of ​​interaction in the symbiotic biological search algorithm and proposes a mutation operator based on the symbiotic strategy, such as formula (8):

[0086]

[0087] Where X best represents the optimal solution, X i represents the i-th solution selected, X i(new) represents the new solution generated, BF is the numerator of the profit, MV represents the relationship between solutions i and j, and the value of MV can be calculated by formula (8); n is the number of elements in the i-th row of the matrix of formula (3). Throughout the text, i, j, and n have the same meaning as in the previous model (formulas 1, 2, and 3).

[0088] Step 7 Recalculate the fitness function value Z, sort from large to small, and save the current optimal solution (the optimal solution is the largest Z value)

[0089] Step 8: Determine whether the maximum number of iterations G has been reached max , output the result; otherwise return to Step 5. Specific embodiments

[0091] In order to verify the feasibility and effectiveness of the present invention, the following example is given: Assume that the number of our weapon platforms is 4, each of which has 25 weapons; the number of incoming targets is 10, and the values ​​of each target are (0.76, 0.88, 0.92, 0.89, 0.74, 0.88, 0.47, 0.66, 0.86, 0.93) respectively; the shooting advantage and target value are shown in Table 1. The parameters are set as: popsize = 100; λ = 0.5; P k =0.05; pd max =1; pd min =0.1; G max =200; F=0.5.

[0092] Table 1 Shooting advantage

[0093]

[0094] Table 2 Optimal weapon target allocation scheme

[0095]

[0096] Table 3 Optimal value / average value / average running time for different population sizes and iteration numbers

[0097]

[0098] Table 1 shows all the weapon allocation schemes finally obtained, and Table 2 shows the optimal value / average value of the present invention and the average running time of multiple runs under different iteration numbers and population sizes. It can be found from the table that the optimal value and the average value are improved to a certain extent with the increase of iteration number and population size. Therefore, in order to increase the solution accuracy, the population size or the number of iterations can be appropriately increased.

[0099] The method of the present invention is compared with the BBODE algorithm proposed by Kumar S et al. in "BBO-DE: Hybrid algorithm based on BBO and DE", (2017 International Conference on Computing, Communication and Automation. IEEE, 2017: 379-383.), the BBOPSODE algorithm proposed by Yaacob et al. in "Hybrid BBO-PSO and higher order spectral features for emotion and stress recognition from natural speech" (Applied Soft Computing, 2017, 56(2): 217-232.), and the Task scheduling algorithm based on improved NSBBO in cloud manufacturing environment by Zheng C et al. (Computer Simulation, 2017, 56(2): 217-232.). Engineering, 2019, 45(10): 26-32.), Zhang Xinming et al.'s "Biogeographic Optimization Algorithm of Differential Migration and Trending Variation" (Small Microcomputer Systems, 2018, 39(06): 1168-1177.) and the basic BBO algorithm are compared. The performance index comparison results of different algorithms are shown in Table 4. The fitness curves of different algorithms are shown in Figure 2 .

[0100] Table 4 Performance comparison of different methods

[0101]

[0102] Depend on Figure 2 It can be seen from Table 4 that the solution accuracy of the method of the present invention is the highest, which is about 5% higher than that of the original BBO algorithm, and the optimal solution obtained is significantly higher than that of several other algorithms. In terms of running time, the running time of the method of the present invention is second only to the basic BBO algorithm, and is significantly higher than that of several other methods, which can meet the real-time requirements of the allocation scheme. However, the optimal solution appears algebraically later. Analysis shows that: the convergence speed of the method of the present invention is slow in the early stage, and the diversity of the population is significantly enhanced by the improved migration and mutation operators, thereby improving the accuracy of the method in the later stage.

[0103] It should be understood that the above specific embodiments are only used to explain the present invention, and are not used to limit the present invention.

Claims

1. A weapon target allocation optimization method based on improved biogeography, characterized in that: The specific steps include: Step 1: Construct a WTA optimization model based on maximizing the cost-effectiveness ratio Suppose there are n weapon platforms, i = 1, 2, ..., n, i is the ordinal number of the weapon platform, and m incoming targets are intercepted, j = 1, 2, ..., m, j is the ordinal number of the incoming target, and the number of weapons equipped on each weapon platform is C i ; V j is the jth target value; Q ij R is the shooting advantage of the i-th weapon platform against the j-th incoming target; ij is the attack risk of the i-th weapon platform on the j-th incoming target; ij The number of weapons assigned to the jth incoming target by the i-th weapon platform is used to establish the multi-objective function as follows: In the formula, Z1 represents the shooting benefit function, Z2 represents the combat cost function, and λ is the weight parameter given by the expert; In order to transform multi-objective optimization into common single-objective optimization, a WTA optimization model based on maximizing the cost-effectiveness ratio is constructed, as shown in formula (2): Where Z is the fitness function value; The constraints in the model have the following meanings: ① Each weapon platform can attack multiple incoming targets at the same time, and multiple weapon platforms can attack the same incoming target, with at least one weapon assigned to each incoming target; ② Each weapon platform is equipped with multiple weapons. The weapons assigned by each weapon platform to the incoming target at the same time cannot exceed the total number of weapons of the weapon platform; Step 2: Set the population size popsize and randomly generate the initial solution of the population. The initial solution is the first randomly generated x ij ; Step 3: Encode the population using an integer-based matrix encoding strategy Let X ij Represents the allocation scheme and uses formula (3) to encode the population: Step 4 Calculate the fitness function value Z according to formula (2), sort them from large to small, and retain the solutions x corresponding to the first k larger fitness values ij ; Step 5: Design a migration operator based on the cosine dynamic adaptive strategy to perform migration operations The specific steps include: Step 5.1 Dynamic selection of habitat migration operators Habitat dynamic selection migration operator: Among them, P k is the probability that the kth solution is selected for migration; popsize is the population size; μ k is the emigration rate of the kth habitat, P k , μ k The a in the upper right corner represents the power of a; μ l is the emigration rate of any l-th habitat, l represents the l-th solution, l = 1, 2, ..., popsize; let a be the selection pressure factor, pd max is the maximum modification probability, pd min is the minimum modification probability; G is the current iteration number, G max is the maximum number of iterations; Step 5.2 Differential mutation migration operator During the Gth iteration, G does not exceed the maximum number of iterations G max , G max Determined according to the specific situation, by randomly selecting SIV features r1, r2, r3 of three different habitats, and satisfying r1≠r2≠r3, a new mutation vector generated by the three features is obtained: X ij =X r1j +F*(X r2j -X r3j ) (6) Where X ij is the solution of the jth dimension of the ith population; X r1j , X r2j , X r3j are the solutions of the jth dimension of the three different populations selected respectively; F is the scaling factor; Step 5.3 Cosine dynamic adaptive selection strategy The cosine dynamic adaptive selection strategy is designed as follows: Where β is the cosine dynamic adaptive selection factor; When the random probability is greater than β, the differential mutation migration operator is used for modification; when the random probability is less than β, the dynamic selection migration operator is used to modify the out-migration location; In the early stage of iteration, the probability of random probability being less than β is relatively high, and the dynamic selection migration operator is used for modification; in the later stage, the probability of random probability being greater than β is relatively high, and the differential mutation migration operator is used for modification; Step 6: Design a mutation operator based on the symbiosis strategy and perform mutation operations In order to improve the quality of the solution, the mutualistic symbiosis idea of ​​the symbiotic biological search algorithm is introduced: for a solution in the ecosystem, another solution in the ecosystem is randomly selected as the symbiotic object, and the favorable factors of each other are absorbed for co-evolution; since the number of solution components in different rows and columns in the matrix encoding of formula (3) is different, and they constrain each other, in order to reflect the co-evolution, the idea of ​​interaction in the symbiotic biological search algorithm is introduced, and a mutation operator based on the symbiotic strategy is proposed, as shown in formula (8): Where X best represents the optimal solution, X i represents the i-th solution selected, X i(new) represents the new solution generated, BF is the profit numerator, MV represents the relationship between solutions i and j, and the value of MV can be calculated by formula (8); n is the number of elements in the i-th row of the matrix of formula (3); Step 7 Recalculate the fitness function value Z, sort from large to small, and save the current optimal solution Step 8: Determine whether the maximum number of iterations G has been reached max , output the result; otherwise return to Step 5.

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