Large-scale multi-objective combinatorial optimization method and its application based on problem reorganization

Abstract

This invention relates to a large-scale multi-objective combinatorial optimization method and its application based on problem restructuring. The method includes the following steps: randomly initializing a population; dividing decision variables into convergent and diversity variables based on decision variable clustering technology; selecting the optimal individual as a new parent population using convergence-oriented directional crossover mutation combined with a convergent environment selection mechanism; when the population encounters selection pressure, reconstructing the problem to transform the large-scale multi-objective optimization into a single-objective optimization; selecting a new parent population using diversity-oriented directional crossover mutation combined with a diversity environment selection mechanism; and outputting the parent population as the optimal solution set for the optimization objective after the termination condition is met. Compared with existing technologies, the complementary search strategy and problem restructuring strategy proposed in this invention can solve large-scale multi-objective optimization problems, handle convergence and diversity issues separately at different optimization stages, and avoid getting trapped in local optima.

Description

Technical Field

[0001] This invention relates to the field of multi-objective evolutionary computation, and in particular to a large-scale multi-objective combinatorial evolutionary optimization method based on problem recombination. Background Technology

[0002] Over the past two decades, various multi-objective evolutionary algorithms have emerged, including Pareto-based Multi-Objective Evolutionary Algorithms (MOEAs), decomposition-based MOEAs, and index-based MOEAs. While most existing MOEAs demonstrate excellent performance in solving multi-objective problems with a small number of decision variables, their performance drops sharply when solving large-scale multi-objective optimization problems (MOPs) with hundreds or even thousands of decision variables (LSMOPs). As the number of decision variables increases linearly, the size (and complexity) of the search space grows exponentially, leading to premature convergence to local optima or to extremely large regions. In recent years, academia and industry have proposed various MOEA frameworks for optimizing LSMOPs, including those based on co-evolution, decision variable analysis, and problem transformation. However, many problems remain to be solved, mainly in the following aspects:

[0003] LSMOEAs based on the co-evolutionary framework require significant time to analyze and group decision variables. Furthermore, improper grouping can lead to correlations between subproblems, necessitating iterative optimization and severely degrading algorithm performance. It's worth noting that the separability assumption between decision variables is not always valid. Therefore, this algorithm has limitations and is unsuitable for solving large-scale MOPs involving interactions between decision variables.

[0004] While LSMOEAs based on decision variable analysis can reduce the problem size to some extent by classifying decision variables, they generate fewer categories (convergent variables, diverse variables, and mixed variables), and the subproblems decomposed may still be large-scale problems. The overall search efficiency of the algorithm still needs to be improved.

[0005] Problem-transformation-based LSMOEAs require finding a problem transformation function to ensure minimal information loss after the original problem is transformed into a new one. However, finding a perfect problem transformation function is extremely difficult, and virtually impossible for particularly complex problems. Furthermore, since each weight corresponds to a set of decision variables, the search of the decision space is incomplete, leading to room for improvement in the quality of the final solution.

[0006] Chinese patent application CN114819040A discloses a dual-population co-evolution method based on dual search, but this method cannot handle multi-objective optimization problems with 500 or even 1000 or more decision variables. Summary of the Invention

[0007] The purpose of this invention is to overcome the shortcomings of the existing technology in that it is difficult to achieve a balance between convergence and diversity in the optimization process, and to provide a large-scale multi-objective combinatorial optimization method based on problem recombination.

[0008] The objective of this invention can be achieved through the following technical solutions:

[0009] As a first aspect of the present invention, a large-scale multi-objective combinatorial optimization method based on problem recombination is provided. The method includes a convergence optimization stage and a diversity optimization stage, and the specific steps include:

[0010] Randomly initialize the population P;

[0011] Based on decision variable clustering technology, decision variables are divided into convergent variables and diversity variables;

[0012] Convergence optimization phase:

[0013] During the evolutionary process, convergent directional crossover mutation is adopted, combined with a convergent environmental selection mechanism, to select the optimal individual as the parent population P′;

[0014] When the population encounters selection pressure, the problem is restructured, transforming large-scale multi-objective optimization into single-objective optimization, and the optimal individual is reselected as the parent population P″.

[0015] Diversity optimization phase:

[0016] During the evolutionary process, directional crossover mutation oriented towards diversity is adopted, and combined with the diversity environment selection mechanism, a population is selected as a new parent population.

[0017] After the termination condition is met, the parent population is output as the optimal solution set as the optimization objective.

[0018] Furthermore, the specific evolution steps of the convergence optimization phase are as follows:

[0019] Using convergence-based binary competition selection, and leveraging crossover and mutation operators, individuals are selected from the current population to generate N offspring Q;

[0020] Calculate the convergence of each individual in the union of the current population P and offspring Q, and select the N best individuals from the union as the new parent population P′;

[0021] Repeat the above process until the termination condition is met;

[0022] Problem restructuring is performed, and large-scale multi-objective optimization is transformed into single-objective optimization through bidirectional weight vector association;

[0023] Producing offspring population A through differential evolution;

[0024] An environment selection mechanism based on convergence strategy selects the optimal N individuals from the union of the offspring population A and the weight vector population Q′ as the new parent population P″.

[0025] Furthermore, the convergence-oriented crossover is performed on a grouping basis of mutually independent convergence variables;

[0026] The directional mutation of convergence involves selecting variables from the convergent variables for mutation.

[0027] Furthermore, the convergence C of solution x d The calculation formula is as follows:

[0028]

[0029] Where M represents the target quantity.

[0030] Furthermore, the problem reorganization strategy specifically involves: reformulating the original large-scale multi-objective optimization problem into a single-objective optimization with relatively small weight variables through bidirectional weight vector association, and using a set of candidate solutions with good convergence and uniform distribution to guide the algorithm to search towards the optimal set.

[0031] Furthermore, the specific evolution steps of the problem recombination strategy are as follows:

[0032] Weight vector association: select r solutions from the current population as a reference solution set, and each reference solution is associated with two direction vectors V. l and V u With two weight variables λ r1 and λ r2 Related;

[0033] Construct a subproblem based on the direction vector and weight variables:

[0034] Z′(Λ)={z 11 (λ 11 ), z 12 (λ 12 ), ..., z r1 (λ r1 )z r2 (λ r2 )}

[0035] Where Λ={λ 11 , λ 12 ,...,λr1 , λ r2} represents the reconstructed decision space;

[0036] The goal space is reconstructed. When the subproblem is reconstructed, the optimization of the decision vector x in the original decision space is transformed into the optimization of the weight vector Λ in the reconstructed decision space. The new optimization problem is expressed as:

[0037] Maximize G(Λ)=H(Z′(Λ))

[0038] Here, H can be any performance metric.

[0039] Furthermore, the specific steps for constructing the subproblem based on the direction vector and weight variables include:

[0040] Calculate the two direction vectors for each reference solution:

[0041] V l =s1-o

[0042] V u =t-s1

[0043] Where s1={x1,...,x d} is the reference solution, and t and o are the upper and lower boundary points of X, respectively;

[0044] Calculate two weight vectors for each reference solution:

[0045]

[0046]

[0047] Where λ11 and λ12 are two weight variables, l max =||to||;

[0048] The expression for constructing the subproblem is:

[0049]

[0050]

[0051] Furthermore, the specific evolutionary steps of the diversity optimization stage are as follows:

[0052] Individuals are selected from the current population P″ using a binary competition selection method based on crowding distance;

[0053] Using the individuals selected above, crossover and mutation are performed only on diversity-related variables to generate N offspring Q″;

[0054] Based on Pareto dominance and crowding distance, half of the population from the union of the current population P″ and offspring Q″ is selected as the new parent population P″′.

[0055] Repeat the above process until the termination condition is met, and output the parent population P″′ as the optimal solution set for the optimization objective.

[0056] Furthermore, the directional crossover of diversity is performed on a per-variability basis, with all diversity variables as the unit of crossover.

[0057] The directional variation of the diversity involves selecting variables from the diversity variables for variation.

[0058] As a second aspect of the present invention, an application of the large-scale multi-objective combinatorial optimization method based on problem reorganization as described above is provided, wherein the application scenarios of the method include resource scheduling and integration optimization of large-scale data centers; the specific application steps include:

[0059] Using indicators including energy consumption, resource loss, data communication traffic, and task completion latency as the problem optimization objectives and the remaining available computing capacity of the server as the constraint, a multi-objective constrained optimization model for the large-scale data center resource scheduling and integration problem is established.

[0060] The number of virtual machines to be scheduled is used as the chromosome length, and the virtual machine number is used as the gene location to encode the individual in the population;

[0061] The population P is initialized based on the encoding method, and iterative optimization of the population is carried out according to any of the above-described large-scale multi-objective combinatorial optimization methods based on problem recombination until the termination condition of the algorithm is met. The output is a Pareto optimal solution set with multiple conflict indicators including energy consumption, resource loss, data communication traffic and task completion delay as optimization objectives.

[0062] Compared with the prior art, the present invention has the following beneficial effects:

[0063] 1) The large-scale multi-objective optimization method provided by this invention, based on a problem reconstruction strategy, can transform a large-scale multi-objective optimization problem into an optimization of a weight vector, thereby solving a large-scale problem.

[0064] 2) Unlike existing large-scale multi-objective evolutionary algorithms that simultaneously maintain population convergence and diversity during a single population iteration optimization process, the complementary search strategy proposed in this invention addresses convergence and diversity issues separately at different optimization stages. In the first stage, the population rapidly approaches the Pareto front, neglecting population diversity. After population convergence, the second stage employs a decision variable clustering method to emphasize population diversity, thereby avoiding getting trapped in local optima.

[0065] 3) Unlike existing multi-objective evolutionary algorithms that use crowding or other methods for environment selection, this invention proposes the concept of convergence to quantitatively determine the convergence of candidate solutions, thereby selecting the solution with better convergence.

[0066] 4) The diversity evolution of this invention can adopt other multi-objective optimization algorithms, thereby designing different instantiation algorithms for different types of multi-objective optimization problems, which has strong scalability. Attached Figure Description

[0067] Figure 1 This is a schematic diagram of the large-scale multi-objective combinatorial evolution method based on problem recombination of the present invention;

[0068] Figure 2 This is a schematic diagram of the approximate Pareto front of the instantiation algorithm of the present invention on the test function LSMOF2;

[0069] Figure 3 This is a schematic diagram of the approximate Pareto front of the instantiation algorithm of the present invention on the test function ZDT1;

[0070] Figure 4 This is an exemplary application flowchart of the large-scale multi-objective combinatorial optimization method provided by the present invention. Detailed Implementation

[0071] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. These embodiments are based on the technical solution of the present invention and provide detailed implementation methods and specific operating procedures. However, the scope of protection of the present invention is not limited to the following embodiments.

[0072] Example 1

[0073] The multi-objective optimization method proposed in this invention addresses convergence and diversity issues at different optimization stages through a complementary search strategy. In the first stage, the population rapidly approaches the Pareto front, neglecting population diversity. When encountering selection pressure, the Pareto optimal set (PS) is directly tracked through problem reconstruction. The algorithm first obtains a set of reference directions in the decision space and associates them with a set of weight variables to locate the PS, then transforms the original large-scale multi-objective optimization problem into a low-dimensional single-objective optimization problem. In the second stage, a multi-objective evolutionary algorithm is used to evenly distribute the optimal solution on the approximate Pareto optimal front. In the first stage, the population rapidly approaches the Pareto front, neglecting population diversity. After population convergence, the second stage employs a decision variable clustering method to emphasize population diversity, thereby avoiding getting trapped in local optima.

[0074] like Figure 1The diagram illustrates a large-scale multi-objective combinatorial optimization method based on problem reorganization. This framework is divided into two stages for population optimization: a convergence-oriented stage (CS) and a diversity-oriented stage (DS). The convergence-oriented stage (CS) first employs a binary competition selection based on convergence degree, utilizing crossover and mutation operators to select individuals from the current population P to generate N offspring Q, where N is the population size. Then, the convergence degree of the union of P and Q is determined by equation (1). Finally, the optimal N individuals are selected from the union as the new parent population P. ′ When the population encounters selection pressure, a problem reconstruction approach is adopted, maintaining a set of weight vectors to transform large-scale multi-objective optimization into single-objective optimization. Offspring population A is produced through differential evolution, and an environmental selection mechanism based on convergence strategy selects the optimal N individuals from the union of offspring population A and the weight vector population Q′ as a new parent population P″, further improving the population's convergence. In the diversity optimization (DS) stage, individuals are selected from the current population P″ using a binary competition selection based on crowding distance. Then, the selected individuals are used to perform crossover and mutation only on diversity-related variables to generate offspring Q″. Finally, based on Pareto advantage and crowding distance, half of the population is selected from the union of parent P″ and offspring Q″ as a new parent population. The specific evolutionary steps of the framework are as follows:

[0075] Randomly initialize the population P with a population size of N;

[0076] Based on decision variable clustering technology, decision variables are divided into two categories (convergent variables and diversity variables).

[0077] Using convergence-based binary competition selection, crossover and mutation operators are used to select individuals from the current population P to generate N offspring Q, where N is the population size;

[0078] Calculate the convergence of each individual in P and Q, and select the best N individuals from the union set as the new parent population P′; the convergence Cd of solution x is calculated using the following formula:

[0079]

[0080] Here, M represents the number of objectives. Convergence is defined as the sum of the objective values ​​of x. For minimizing MOP, the smaller the value of Cd, the better the convergence of the solution x.

[0081] After the termination condition is met, a problem-based reconstruction strategy is adopted to start from the current population P. ′ We select r solutions as the reference solution set. Then, we use Equation 2 to calculate the two direction vectors for each reference solution.

[0082] Vl l =s1-o

[0083] Vu u =t-s1 (2)

[0084] Where s1={x1,…,x d} is the reference solution, and o and t are the upper and lower boundary points of X, respectively.

[0085] Use Formula 3 to calculate the two weight vectors for each reference solution.

[0086]

[0087]

[0088] Where, λ 11 and λ 12 There are two weight variables, l max =||to||.

[0089] By selecting a reference solution set of size r, once each reference solution is associated with two direction vectors and two weight variables, a total of 2r subproblems can be constructed.

[0090]

[0091]

[0092] The subproblem is Z′(∧)={z 11 (λ 11 ), z 12 (λ 12 ), ..., z r1 (λ r1 )z r2 (λ r2 )}, where ∧={λ 11 , λ 12 , …, λ r1 , λ r2} represents the reconstructed decision space.

[0093] Once the subproblem is restructured, optimizing the decision vector x in the original decision space becomes optimizing the weight vector. Correspondingly, the objective space can be reduced, and the new optimization problem can be reformulated as follows:

[0094] maximize G(∧)=H(Z′(∧)) (5)

[0095] Here, H can be any performance metric.

[0096] The offspring population A is produced through differential evolution, and the optimal N individuals are selected from the union of A and Q′ as the new parent population P″ based on the environmental selection mechanism of the convergence strategy.

[0097] After the termination condition is met, the process proceeds to Data Sorting (DS). First, a binary competition selection based on crowding distance is used to select individuals from the current population P″. Then, using the selected individuals, crossover and mutation are performed only on diversity-related variables to generate N offspring Q″. Finally, based on Pareto advantage and crowding distance, half of the populations from the union of P″ and Q″ are selected as the new parent population P″′. After the termination condition is met, the parent population P″′ is output as the optimal solution set for the optimization objective.

[0098] 1) CS employs convergence-oriented directional crossover mutation during evolution, combined with convergence-oriented environmental selection mechanisms, to maintain the convergence of the population;

[0099] 2) DS employs directional crossover variation oriented towards diversity during evolution, combined with diverse environmental selection mechanisms to maintain population diversity.

[0100] Preferably, the aforementioned convergence and diversity crossover and mutation operators are implemented through convergence calculation and decision variable analysis, respectively, with the specific steps as follows:

[0101] 1) Clustering of decision variables: This method uses k-means clustering to divide variables into two categories (convergent variables and diversity variables).

[0102] 2) Classification of convergence variables: The formula for calculating the convergence degree Cd of solution x is shown in (1).

[0103] 3) Convergent crossover is performed on a group of mutually independent convergent variables; diversity crossover is performed on a group of all diversity variables.

[0104] 4) Convergent variation selects variables from convergent variables for variation; diversity variation selects variables from diversity variables for variation.

[0105] The specific problem reconstruction strategies used by CS during the evolution process are as follows:

[0106] Problem Restructuring Strategy: This strategy addresses the choice pressure caused by large-scale decision variables. Specifically, it consists of three steps: First, weight vector association: select r solutions from the current population P as a reference solution set, with each reference solution associated with two direction vectors and two weight variables. Second, subproblem construction: construct the subproblem Z′(∧)={z... 11 (λ 11 ), z 12 (λ 12 ), ..., zr1 (λ r1 )z r2 (λ r2 )}, where ∧={λ 11 , λ 12 , …, λ r1 , λ r2 Let} represent the reconstructed decision space. The third step is to reconstruct the target space. Once the subproblem is reconstructed, the optimization of the decision vector x in the original decision space is transformed into the optimization of the weight vector ∧ in the reconstructed decision space. The new optimization problem can be expressed as Equation 5.

[0107] The diversity evolution stage can employ other multi-objective optimization algorithms, thereby designing different instantiation algorithms for different types of multi-objective optimization problems, which has strong scalability.

[0108] Table 1 shows the mean IGD values ​​obtained by the two multi-objective evolutionary algorithms on nine large-scale test cases. To reduce the impact of random errors on the calculation results, each algorithm was run independently 30 times in each test case in this embodiment, and the mean IGD index obtained by each algorithm on each test case was calculated. The IGD index measures the distance between the true Pareto front and the approximate Pareto front obtained by the algorithm. Generally, the smaller the IGD index value, the better the convergence and diversity of the algorithm.

[0109] Table 1

[0110] Problem M D The algorithm instantiated by this framework LSMOF LSMOP1 2 1000 2.6390e-1(1.63e-2)+ 6.2856e-1(2.77e-2) LSMOP2 2 1000 1.1624e-2(3.74e-4)+ 1.8858e-2(3.47e-4) LSMOP3 2 1000 1.5680e+0(1.61e-3)+ 1.5729e+0(4.09e-4) LSMOP4 2 1000 2.5755e-2(7.90e-4)+ 3.8112e-2(1.95e-3) LSMOP5 2 1000 6.8558e-1(4.40e-2)+ 7.4209e-1(3.39e-16) LSMOP6 2 1000 5.6957e-1(1.62e-1)- 3.1236e-1(4.03e-4) LSMOP7 2 1000 1.5087e+0(4.30e-4)- 1.5083e+0(7.70e-4) LSMOP8 2 1000 5.3281e-1(2.17e-1)+ 7.4209e-1(3.39e-16) LSMOP9 2 1000 5.1013e-1(4.32e-3)+ 8.0686e-1(8.94e-4)

[0111] Figure 2 The method of this invention obtains an approximate Pareto front on a 2-objective, 1000-decision-variable LSMOP2 benchmark function. Figure 3 The method of this invention obtains an approximate Pareto front on a ZDT1 benchmark function with 2 objectives and 1000 decision variables.

[0112] Example 2

[0113] As a second aspect of the present invention, an application of the method described in the above embodiments is provided. A typical application scenario of this method is resource scheduling and integration optimization in large-scale data centers. Figure 4 The specific application process is shown below:

[0114] Step 1: First, using indicators such as energy consumption, resource loss, data communication traffic, and task completion latency as the problem optimization objectives, and the remaining available computing capacity of the server as the constraint, a multi-objective constrained optimization model for the large-scale data center resource scheduling and integration problem is established.

[0115] Step 2: Then, using the number of virtual machines to be scheduled as the chromosome length and the virtual machine number as the gene location, the individual in the population is encoded;

[0116] Step 3: Finally, initialize the population P based on the encoding method, and follow... Figure 1 The large-scale multi-objective combinatorial optimization method based on problem recombination is shown to iteratively optimize the population until the algorithm's termination condition is met, and outputs the Pareto optimal solution set with multiple conflicting indicators such as energy consumption, resource loss, data communication traffic, and task completion delay as optimization objectives.

[0117] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A large-scale multi-objective combinatorial optimization method based on problem recombination, characterized in that, The application scenarios of the method include resource scheduling and integration optimization in large-scale data centers; Using indicators including energy consumption, resource depletion, data communication traffic, and task completion latency as the problem optimization objectives, and the remaining available computing capacity of the server as the constraint, a multi-objective constrained optimization model is established for the large-scale data center resource scheduling and integration problem; the number of virtual machines to be scheduled is the chromosome length, and the virtual machine number is the gene position, encoding the individual in the population. The population P is randomly initialized, and iterative optimization of the population is performed, including a convergence optimization phase and a diversity optimization phase. The specific steps are as follows: Based on decision variable clustering technology, decision variables are divided into convergent variables and diversity variables; Convergence optimization phase: During the evolutionary process, convergent directional crossover mutation is employed, combined with a convergent environmental selection mechanism, to select the optimal individuals as parental populations. ; When a population encounters selection pressure, the problem is restructured, transforming large-scale multi-objective optimization into single-objective optimization, and the optimal individual is reselected as the parent population. The evolutionary steps of the problem reconstruction are as follows: Weight vector association: select r solutions from the current population as a reference solution set, each reference solution is associated with two direction vectors. and With two weight variables and Related; Construct a subproblem based on the direction vector and weight variables: in, For the reconstructed decision space; The goal space is reconstructed. When the subproblems are reconstructed, the optimization of the decision vector x in the original decision space is transformed into the optimization of the weight vector. In the reconstructed decision space, the new optimization problem is formulated as follows: Where H represents any performance metric; The specific steps for constructing the sub-problem based on the direction vector and weight variables include: Calculate the two direction vectors for each reference solution: in, For reference, , These are the upper and lower boundary points of X, respectively; Calculate two weight vectors for each reference solution: Among them, 𝜆11 and 𝜆12 are two weight variables. ; The expression for constructing the subproblem is: Diversity optimization phase: During the evolutionary process, directional crossover mutation oriented towards diversity is adopted, and combined with the diversity environment selection mechanism, a population is selected as a new parent population. After the termination condition is met, the parent population is output as the optimal solution set with multiple conflicting indicators such as energy consumption, resource loss, data communication traffic and task completion delay as optimization objectives.

2. The large-scale multi-objective combinatorial optimization method based on problem recombination according to claim 1, characterized in that, The specific evolutionary steps of the convergence optimization phase are as follows: Using convergence-based binary competition selection, and leveraging crossover and mutation operators, individuals are selected from the current population to generate N offspring Q; Calculate the convergence of each individual in the union of the current population P and its offspring Q, and select the N optimal individuals from the union as the new parent population. ; Repeat the above process until the termination condition is met; Problem restructuring is performed, and large-scale multi-objective optimization is transformed into single-objective optimization through bidirectional weight vector association; Producing offspring population A through differential evolution; An environment selection mechanism based on a convergence strategy selects from the offspring population A and the weight vector population. The best N individuals were selected as the new parent population. .

3. The large-scale multi-objective combinatorial optimization method based on problem recombination according to claim 2, characterized in that, The convergence-oriented crossover is performed on a grouping basis of mutually independent convergence variables; The directional mutation of convergence involves selecting variables from the convergent variables for mutation.

4. The large-scale multi-objective combinatorial optimization method based on problem recombination according to claim 2, characterized in that, The convergence of solution x The calculation formula is as follows: in, Indicates the target quantity.

5. The large-scale multi-objective combinatorial optimization method based on problem reorganization according to claim 1, characterized in that, The problem reconstruction specifically involves: by using bidirectional weight vector association, the original large-scale multi-objective optimization problem is reformulated into a single-objective optimization with relatively small weight variables, and a set of candidate solutions with good convergence and uniform distribution is used to guide the algorithm to search towards the optimal set.

6. The large-scale multi-objective combinatorial optimization method based on problem reorganization according to claim 1, characterized in that, The specific evolutionary steps of the diversity optimization stage are as follows: Employing a binary competition based on crowding distance to select from the current group Select an individual; Using the individuals selected above, crossover and mutation are performed only on diversity-related variables to generate N offspring. ; Based on Pareto advantage and crowd distance, from the current group and offspring And select half of the population as the new parent population. ; Repeat the above process until the termination condition is met, and output the parent population. The optimal solution set is the objective of optimization.

7. The large-scale multi-objective combinatorial optimization method based on problem recombination according to claim 6, characterized in that, The directional crossover of diversity is performed on a per-variability basis, with all diversity variables as the unit. The directional variation of the diversity involves selecting variables from the diversity variables for variation.

Images