# A Multi-Offspring Genetic Algorithm Based on Sorting Grouping Selection and Combination Pairing Crossover

Authors: Xin Jin; Fulin Wang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

A multi-offspring genetic algorithm based on sorting grouping selection and combination pairing crossover was proposed in this paper. First, individuals in the population were sorted according to their objective function values, and the remaining individuals were divided into several groups after eliminating the worst. Then, the selection operation, which has the advantage of a simplified calculation that can be easily performed, was implemented. Second, a combination pairing crossover operator was designed. Individuals from different groups were selected for new combinations. In a combination, if a crossover condition was met, individuals pairing and transposons crossover operation were made. Otherwise, offspring copies of the individuals in the combination were used, which increase the opportunity of generating better offspring by producing multi-offspring. Finally, a new population was formed by adopting basic bit mutation operator, elitism policy and the strategy of multi-offspring population competition. Moreover, the catastrophe mechanism has been introduced into improved algorithm to avoid premature convergence. The test results on the functions of CEC 2017 test suites shown that the algorithm proposed in this paper has better search performance, stability and faster convergence to the global optimal solution. These results thus verified the effectiveness and feasibility of the algorithm proposed in this paper. Applying the improved algorithm to the location optimization problem of agricultural product logistics facilities, it shown that the improved algorithm is an effective method to solve the location optimization.


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## Body

## 1. Introduction

The Genetic Algorithm (GA) was first reported in 1975 by Holland et al. [1–3]. Due to its strong ability to perform global searching, the GA was widely used in many fields such as function optimization [4], production scheduling [5, 6], signal processing [7], and artificial intelligence [8]. Nevertheless, the GA has some shortcomings in the process of its application. For instance, the GA has poor local searching, premature converging, and slow convergence speed [9]. To overcome these shortcomings, many scholars have suggested innovative ideas to improve and apply the GA over the past four decades. Research on the GA can be grouped into the following categories:(1)
Optimizing gene coding: Some new coding types such as the chaos gray-coded GA [10] and matrix coding GA [11] have been proposed, and the influence of different coding types on the performance of the GA has been studied [12].(2)
Improving parameters: Many new ideas have been presented, such as the use of a fuzzy controller to adjust the cross-probability parameters to avoid a premature phenomenon [13]; an adaptive genetic algorithm was presented as an intelligent algorithm, of which the probability of crossover and mutation was dynamically adjusted according to the individual’s fitness value [14], the co-evolution approach was used to adjust the crossover and mutation probability, which is able to accelerate the convergence and prevent the prematurity [15], an IAGA optimization model was presented, of which the crossover probability is adaptively adjusted by introducing a nonlinear function and the mutation probability is adaptively adjusted using complementary idea [16].(3)
Improving the operators: The selection operator, such as the kin selection operator [17], probabilistically selection operator [18], and back controlled selection operator [19] have been improved. The crossover operator is an important operator that affects the global search of the genetic algorithm; thus, many improved genetic algorithms have been proposed, including GAs based on the phenotype crossover operator [20], chaotic crossover operator [21], fuzzy crossover operator [22] and others. To enhance the local search capability of the GA, some mutation operators have been designed, including the greedy sub tour mutation [23] and asymmetric mutation operator [24].(4)
Combining the GA with other algorithms: Hybrid genetic algorithms can make full use of the advantages of different algorithms, including the simulated annealing genetic algorithm [25], niche genetic algorithm [26], chaos genetic algorithm [27], and quantum genetic algorithm [28]. In recent years, some representative evolutionary computation methods have emerged such as self-adaptive artificial bee colony algorithm [29], return-cost-based binary firefly algorithm [30], ensemble discrete differential evolution algorithm [31], self-adaptive differential evolution algorithm [32], mixed-variable differential evolution approach [33], hybrid discrete water wave optimization algorithm [34] and so on, which also make it possible to study the hybrid algorithm of genetic algorithm and these representative evolutionary algorithms.(5)
Application research of the GA: Applications of the GA in the simultaneous identification of atmospheric pollution sources [35], mobile robot path planning [36], multi-objective analog/RF circuit optimization [37], and cloud task scheduling [38] have been studied.These algorithm researches have improved the computational speed and promoted the development of GA theory with new applications. A common feature of these improved GAs is that the number of offspring generated is the same as the number of parents. However, the basic viewpoints of Darwin’s natural selection theory are explosive reproduction, survival struggle, genetic variation and survival of the fittest in the process of biological evolution. Explosive reproduction refers to mass reproduction in a short time, that is, it can produce a large number of offspring in a certain time. In fact, most organisms have the tendency of over reproduction. Numerous offspring realize the survival of the fittest through their own genetic variation and mutual survival competition. Multi-offspring genetic algorithm was proposed based on this theory. In 2015, Wang et al. provided a multi-offspring genetic algorithm (MOGA) based on single-point crossover [39]. In 2016, Wang et al. studied the MOGA based on two-point crossover [40]. On the basis of above two studies, MOGA has been applied to optimize the BP neural network [41], solving the traveling salesman problem [42] and take-out delivery problem [43]. The above-mentioned research results indicate that MOGA indeed generated improved results compared to the simple genetic algorithm. MOGA can increase the excellent individuals survival and diversity of the species by generating more offspring than the number of parents [40].In the existing literature regarding MOGA, the MOGA generates multioffspring by means of roulette wheel selection and single-point crossover or two-point crossover. Roulette wheel selection operator used to randomly select individuals according to individual fitness. Although it can increase the selection probability of excellent individuals, it affects the diversity of the population. Regardless of which above crossover operation is performed in MOGA, multi-offspring are generated by increasing crossover operation times in the same pair of parents. The greater crossover operation times adopted in the same two parents, the more the offspring generated. As a result, the generated individuals will be more similar, resulting in premature local convergence.For the sake of competitiveness and diversity of population, and reaching better ability of searching optimum and better effect of avoiding premature convergence than MOGA, the multi-offspring genetic algorithm based on sorting grouping selection and combination pairing crossover (SGMOGA) was proposed in this paper. Different from traditional MOGA, SGMOGA used easily implemented sorting grouping selection operator to select individuals from each group can avoid inbreeding and ensure population diversity. The multi-offspring generated by combination paring crossover not by increasing crossover operation times among a pair of parents to avoid premature convergence. The transposons crossover operator can transfer the good genes of excellent individuals in the crossover, to speed up the convergence. In order to avoid falling into local convergence, catastrophe mechanism also has been introduced into this algorithm to make the population restart diversified search when trapped into local optimum, to find the global optimal solution. The experimental results with CEC 2017 test suites shown that the SGMOGA was more effective and easier to implement than the MOGA introduced in the above-mentioned literature.The major contributions of this paper are as follows: (1) According to the biological theory, this paper designed a multi-offspring genetic algorithm based on sorting grouping selection and combination pairing crossover; (2) Proposing the sorting grouping selection operator, combination pairing transposons crossover operator, population competition strategy and catastrophe mechanism to simplify the calculation, maintain the diversity of population, avoid premature convergence and increase the probability of excellent individuals in offspring population, which makes the algorithm not only have faster operation speed, but also have better convergence; (3) Our proposed SGMOGA has been compared with some comparison algorithm, using the boxplot andT-test to prove SGMOGA has significant statistical performance advantages over the other algorithms; (4) Investigating the effects of GA parameters such as population size, crossover and mutation values on the performance of the proposed method.


## 2. The Multioffspring Genetic Algorithm Based on Sorting Grouping Selection and Combination Pairing Crossover

The SGMOGA proposed in this paper includes elements such as parameter encoding, generation of an initial population, the sorting grouping selection operator, the combination pairing transposons crossover operator, the mutation operator, catastrophe mechanism and evolutionary strategy. The SGMOGA does binary calculations, and the other elements are as follows:


### 2.1. Generating the Initial Population

Randomly generating the initial population: to keep comparability among algorithms, it should ensure that the same initial population is called in the same calculations. The different algorithms can be compared by the average values of their operation results because each algorithm should be repeatedN times. Taking this reality into consideration, N initial populations are generated randomly, each population size n. The ordering of them is fixed, and they are called in the same sequence in every algorithm.


### 2.2. Sorting Grouping Selection Operator

#### 2.2.1. Sorting of Individuals

Initially, the objective function values of each individual are calculated. Then, all individuals in the population are sorted according to the results of the calculation. There are two cases to consider: in one case, all individuals are sorted in descending order on the basis of objective function values in the maximization problem, and in the other case, all individuals are sorted in ascending order in the minimization problem. The purpose of such sorting is to ensure that the best individual always comes first, and the worst individual comes last, regardless of whether the objective function is to find the maximum or the minimum value. The SGMOGA reduces the operating time to some extent because the algorithm directly calculates the objective function value and then sorts individuals without fitness calculation processing, which most GAs and MOGAs include.


#### 2.2.2. Eliminate Worse Individuals and Divide the Remainder into Groups

In the biological world, those who are less competitive may not have the opportunity to participate in reproduction. Based on this, some worse individuals may be eliminated, and the remainder are parents to breed. After individuals sorting, the number of groups needs to be determined according to the quantity of offspring to be generated. Then, the parent individuals are distributed equally among groups. To facilitate the subsequent SGMOGA operation and ensure that each remaining individual can participate in the combination, the number of individuals in each group must be equal. This involves the determination of the initial population size and whether it is necessary to eliminate worse individuals before grouping and the quantity of individuals to be eliminated. Due to the initial population may be divided into any number of groups such as 2, 3, 4, 5 groups to reach the demands of various experiments, the initial population size should be the least common multiple (60) of the number of groups, in order to ensure that all individuals in the population can be allocated and that the number of individuals in each group is equal without worse individuals were eliminated. Assuming all individuals were divided into 2∼10 groups, the initial population size should be set to 2520 (the least common multiple of 2∼10). However, when the initial population is excessively large, with an increase in the number of groups, the population size of the offspring will be rapidly expanding, which will affect the operation speed of the GA. For the sake of reasonably controlling the operation speed of the algorithm, the initial population size should be set to an appropriate value. Then, a certain number of worse individuals are eliminated properly to ensure that the number of individuals in each group is equal in the case of any number of groups.For instance, if the population size is 10 and the number of groups is 3, all individuals in the population will be sorted according to the objective function value, and the last individual will be eliminated in order to ensure that the number of individuals each group is equal. The remaining 9 individuals will be equally allocated to 3 groups. The individuals ranking in the first 3 belong to the first group, the individuals ranking from 4th to 6th are sorted into the second group, and the last 3 individuals in the sequence are sorted into the third group. The sorting elimination and grouping process is shown in Table1.Table 1
Sorting elimination and grouping process.
Before sortingAfter sortingThe first groupThe second groupThe third groupM1M1′M1′M4′M7′M2M2′M2′M5′M8′M3M3′M3′M6′M9′M4M4′M5M5′M6M6′M7M7′M8M8′M9M9′M10M10′ ⟶ eliminatedWhen the initial population size isn and the number of groups is k, the number of individuals eliminated and the remainders can be assessed by (1) and (2):(1)z=n−n/kk,(2)r=n−z=n/kk,where z is the number of individuals eliminated, r is the number of remainders, n is the initial population size, and k is the number of groups.  is the rounding down symbol.After sorting and elimination, the remainders are divided into groups. The number of individuals in each group is as shown in :(3)m=r/k,where m is the number of individuals in each group.We assume that the initial population size is 100, and the number of groups is a value from 2∼10. Then, we can calculate the number of individuals eliminated and the remainders by (1) and (2), and we can calculate the number of individuals in each group by equation (3). The calculation results are shown in Table 2.Table 2
Number of individuals eliminated and remainders and the quantity of individuals in each group under different number of groups.
kzrm20100503199334010025501002064961672981484961291991110010010


#### 2.2.3. Selection Operator

After sorting and grouping of individuals in the population, it is necessary to select individuals from each group. Selecting individuals with the same ranking position in each group due to the obvious divergence among the individuals in different groups after sorting and grouping. This selection operation is to avoid the randomness caused by roulette selection, and to ensure the diversity of the population to a certain extent because of the differences between individuals which are used to crossover. In the case of three groups as Table1, M1′M4′M7′ were selected at the first selection, M2′M5′M8′ were selected at the second selection, and M3′M6′M9′ were selected at the third selection. Their ranking positions in each group are 1, 2 and 3 respectively, at same intervals of individuals each selection, which can ensure that there are certain differences between individuals in each selection.


### 2.3. Combination Pairing Crossover Operator

#### 2.3.1. Combination and Pairing

One individual from each group is chosen at a time, and they are combined to form new combinations. By makingm choices in each group with all groups having the same number of members, m combinations are obtained. The above combination operation is performed to ensure the diversity of the combination to a certain extent because of the differences among individuals in combinations. Then, individuals in new combinations can make pairing and crossover if it meets the crossover condition. We assume that the crossover condition is met, and the process of combination and pairing is shown in Table 3.Table 3
Combination and pairing process.
NumberCombinationPairing1M1′M4′M7′M1′ ↔ M4′M1′ ↔ M7′M4′ ↔ M7′2M2′M5′M8′M2′ ↔ M5′M2′ ↔ M8′M5′ ↔ M8′3M3′M6′M9′M3′ ↔ M6′M3′ ↔ M9′M6′ ↔ M9′Subsequently, the relationship between the number of new combinations, the quantity of individuals in the combination and the number of groups should be discussed. According to Tables1–3, it can be seen that the number of new combinations is equal to the number of individuals in each group, while the number of individuals in the new combination is equal to the number of groups. Thus, the number of new combinations and the quantity of individuals in each combination can be expressed by (4) and (5):(4)c=m=r/k,(5)w=k,where c is the number of new combinations, and w is the number of individuals in the new combination.In the combination, any two individuals can make a pair. Thus, the number of pairs in each combination that containsw individuals can be expressed by :(6)pairs=Cw2,where pairs is the number of pairs in each combination.If we assume that the initial population size is 100 and that the number of groups is a value from 2∼10, we can calculate the number of new combinations by (4), the number of individuals in each new combination by equation (5) and the number of pairs in each combination by equation (6). The result is shown in Table 4.Table 4
Number of individuals and pairs in each combination and the number of combinations.
kcwpairs25021333334254652051061661571472181282891193610101045


#### 2.3.2. Transposons Crossover Operator

To increase the convergence speed of the global optimal solution, a transposons crossover operator was designed in this paper that emulates the gene transposition in the genome that was discovered by Nobel Laureate, Barbara McClintock, in her work on the corn plants. Transposons are also known as jumping genes, McClintock’s experimental observation indicated that there were two ways in which the transposons could jump around the genome. Two jumping ways were called cut-and-paste and copy-and-paste [44], as depicted in Figures 1 and 2, respectively. In both figures, the gray part of the chromosome represents transposons.Figure 1
Cut-and-paste transposons. (a) Cut-and-paste transposons—same chromosome (b) cut-and-paste transposons—different chromosomes.
(a)(b)Figure 2
Copy-and-paste transposons. (a) Copy-and-paste transposons—same chromosome (b) copy-and-paste transposons—different chromosomes.
(a)(b)The cut-and-paste operation is that the transposons is cut from an original position of a chromosome and pasted it into a new position. As for the copy-and-paste operation, the transposons replicates itself and the copy is pasted into a new area of the chromosome, while the original one remains unchanged [44].Based on the transposons copy-and-paste operation between different chromosomes, a transposons crossover operator was proposed in this paper. First, a condition of crossover between a pair individual was set. Paired individuals in combinations carry out the crossover operation if the random number generated was less than the crossover probabilityPc, otherwise the paired individuals were retained as offspring. When the crossover condition was met, paired individuals in combinations carry out the crossover operation and generate multi-offspring.Six offspring should be generated by the crossover or copy operation when there are three individuals in a combination. The process of the crossover or copy operation in the first combination shown in Table3 is shown in Table 5.Table 5
Process of the crossover or copy operation.
CombinationConditionOperationPairingOffspringM1′M4′M7′rand<PcCrossoverM1′,M4′child1, child2M1′,M7′child3, child4M4′,M7′child5, child6rand>PcCopyM1′ ⟶ child1, child2M4′ ⟶ child3, child4M7′ ⟶ child5, child6To compare with the MOGA, the following discussion concerns the offspring generation process by transposons crossover operator between paired individuals in a combination.According to equation (5) and (6), when the initial population was divided into k groups, each new combination contain k parents. Ck2 pairs of parents will be formed when the crossover condition was met. In order to preserve and transmit the excellent genes of excellent individuals to speed up the convergence of the algorithm, we set that the transpositions length of each pair of crossover individuals is different, that is, the transpositions length of excellent individuals in each pair of individuals is greater than that of poor individuals. Transposons crossover operator was performed each pair of individuals, firstly, the position of transposons each individual and the insertion position of the transposons in another individual should be determined. Then, the transposons of one individual was inserted into the insertion position of another individual in the paired individual. Finally, the transposons crossover operation was completed after the transposons in the paired individuals were exchanged. It was assumed that the initial population was divided into three groups, and the three parents in a combination were represented as follows:parent1 = 110011110001000parent2 = 001111001001011parent3 = 010110011010011According to the above-mentioned three individuals, they can form three pairs of parents. In the first pair of individuals composed of parent 1 and parent 2, their transposons were gene strings located between 4 and 11, 6 and 10, respectively, the insertion position were 9 and 2. child1 and child2 were generated by exchanging transposons and inserting them into the insertion position of another parent individual. The other two pairs of parent individuals in the combination also used the same transposons crossover operation to produce child3, child4, child5 and child6. The process of multi-offspring generation by the transposons crossover operator was shown in Figure3.Figure 3
Process of multi-offspring generation by transposons crossover operator.We observed that the number of multioffspring generated by SGMOGA is mainly achieved by sorting grouping selection and combination pairing crossover, which is not directly related to crossover operation times among a pair of parents. Sorting grouping selection and the combination pairing crossover operator can increase the convergence speed of the global optimal solution and maintain population diversity.


#### 2.3.3. Relationship between the Number of Groups and Offspring Generated

Different from other MOGAs regarding the manner of offspring production, SGMOGA can generate offspring through introducing the sorting grouping and combination pairing mechanism into the GA, and the number of offspring produced is related to the number of groups and not related to the number of times of crossover operation among a pair of parents as in other MOGAs. Then, the relationship between the number of groups and the quantity of offspring generated will be discussed.SGMOGA can generate two offspring by crossover in one pair of parents. Therefore, the number of offspring is twice the number of parent pairs. The total number of offspring generated by the initial population can be expressed by:(7)o=2⋅pairs⋅c,where o is the total number of offspring.Because all offspring are generated byr parents, the quantitative proportional relationship between o offspring and r parents can be assessed by :(8)prop=or=2⋅pairs⋅cr=2⋅Cw2⋅r/kr=k−1,where prop is the proportion between o offspring and r parents.As previously assumed, the relationship between the number of groups and the quantity of offspring is shown in Table6.Table 6
The relationship between the number of groups and the quantity of offspring.
korprop21001001319899243001003540010046480965758898686729679792998109001009As shown in Table6, the number of offspring produced by SGMOGA was positively correlated with the number of groups. Specifically, when the number of groups is k, the number of offspring generated is k−1 times of the remaining parents after elimination. The flow chart of sorting grouping selection and the combination pairing crossover operator is shown in Figure 4.Figure 4
The flow chart of sorting grouping selection and the combination pairing crossover operator.


### 2.4. Mutation Operator

The SGMOGA approach proposed in this paper adopt the basic bit mutation operator, that is, a gene bit randomly designated on the binary string is modified with the mutation operator when the generated random number is less than the mutation probability value. The basic bit mutation operator can realize the local search function of the algorithm, which is a common method of a GA under binary coding.


### 2.5. Population Competition

For SGMOGA, it is necessary to keep the population size unchanged; otherwise, the running time of the algorithm will increase significantly with the increase of the number of offspring. Therefore, population competition is introduced into SGMOGA for the sake of maintaining an unchanged parent population size; therefore, the excellent individuals within the population can survive better, which will promote the population to develop in a better direction. In this process,z worse individuals are first eliminated after sorting n individuals according their objective function value; then r individuals are divided into k groups. There will be r parents participating in the combination pairing crossover operation. As previously analyzed, the number of offspring generated after combination pairing crossover is k−1 times as many as r parents, so the number of offspring generated after crossover by SGMOGA is illustrated by :(9)n1=k−1r,where n1 is the number of offspring generated by the crossover of r parents.(2) is substituted into equation (9) and it can be changed into :(10)n1=k−1n/kk.For the sake of population competition,s elitists are picked from the parent population and put into the offspring population. After crossover and the first-time elitist reserved, the offspring population size is shown in :(11)n2=s+n1=s+k−1n/kk,where n2 is the offspring population size, s is the number of the first-time elitist reserved.To avoid the damage of mutation to excellent individuals,s elitists are first picked from n2 individuals, then the mutation operation is performed on n1 individuals generated by crossover. Finally, after mutation and the second-time elitist reserved, there are n2 individuals generated.It is thus clear from (11) and Table 6 that the population size of offspring in SGMOGA is far larger than the initial population size. This indicates a larger size disparity between the size of parents and offspring with an increasing number of groups. In this large population, not only many individuals with poor survivability will exist, but also the operation speed of the algorithm will be seriously affected. Therefore, we need to eliminate some worse individuals through population competition. Consequently, after mutation and the second-time elitist reserved, picking n optimal individuals from n2 offspring, will form a new parent population such that the population size is unchangeable at each iteration, which can speed up the algorithm computation.


### 2.6. Catastrophe Mechanism

Catastrophe is the extinction of a large number of species or individuals due to major catastrophic accidents in biological evolution. Only species or individuals with strong adaptability can survive and reproduce [45]. In order to avoid falling into local convergence, catastrophe mechanism is introduced into this algorithm to make the population restart diversified search, so as to find the global optimal solution. In SGMOGA, when the optimal value of the objective function for three consecutive generations does not change, the catastrophe mechanism start work, which the excellent individual is retained, while some worse are reinitialized when the generated random number is less than the catastrophe probability, to maintain the diversity of the population and restart diversified search.


## 2.1. Generating the Initial Population

Randomly generating the initial population: to keep comparability among algorithms, it should ensure that the same initial population is called in the same calculations. The different algorithms can be compared by the average values of their operation results because each algorithm should be repeatedN times. Taking this reality into consideration, N initial populations are generated randomly, each population size n. The ordering of them is fixed, and they are called in the same sequence in every algorithm.


## 2.2. Sorting Grouping Selection Operator

### 2.2.1. Sorting of Individuals

Initially, the objective function values of each individual are calculated. Then, all individuals in the population are sorted according to the results of the calculation. There are two cases to consider: in one case, all individuals are sorted in descending order on the basis of objective function values in the maximization problem, and in the other case, all individuals are sorted in ascending order in the minimization problem. The purpose of such sorting is to ensure that the best individual always comes first, and the worst individual comes last, regardless of whether the objective function is to find the maximum or the minimum value. The SGMOGA reduces the operating time to some extent because the algorithm directly calculates the objective function value and then sorts individuals without fitness calculation processing, which most GAs and MOGAs include.


### 2.2.2. Eliminate Worse Individuals and Divide the Remainder into Groups

In the biological world, those who are less competitive may not have the opportunity to participate in reproduction. Based on this, some worse individuals may be eliminated, and the remainder are parents to breed. After individuals sorting, the number of groups needs to be determined according to the quantity of offspring to be generated. Then, the parent individuals are distributed equally among groups. To facilitate the subsequent SGMOGA operation and ensure that each remaining individual can participate in the combination, the number of individuals in each group must be equal. This involves the determination of the initial population size and whether it is necessary to eliminate worse individuals before grouping and the quantity of individuals to be eliminated. Due to the initial population may be divided into any number of groups such as 2, 3, 4, 5 groups to reach the demands of various experiments, the initial population size should be the least common multiple (60) of the number of groups, in order to ensure that all individuals in the population can be allocated and that the number of individuals in each group is equal without worse individuals were eliminated. Assuming all individuals were divided into 2∼10 groups, the initial population size should be set to 2520 (the least common multiple of 2∼10). However, when the initial population is excessively large, with an increase in the number of groups, the population size of the offspring will be rapidly expanding, which will affect the operation speed of the GA. For the sake of reasonably controlling the operation speed of the algorithm, the initial population size should be set to an appropriate value. Then, a certain number of worse individuals are eliminated properly to ensure that the number of individuals in each group is equal in the case of any number of groups.For instance, if the population size is 10 and the number of groups is 3, all individuals in the population will be sorted according to the objective function value, and the last individual will be eliminated in order to ensure that the number of individuals each group is equal. The remaining 9 individuals will be equally allocated to 3 groups. The individuals ranking in the first 3 belong to the first group, the individuals ranking from 4th to 6th are sorted into the second group, and the last 3 individuals in the sequence are sorted into the third group. The sorting elimination and grouping process is shown in Table1.Table 1
Sorting elimination and grouping process.
Before sortingAfter sortingThe first groupThe second groupThe third groupM1M1′M1′M4′M7′M2M2′M2′M5′M8′M3M3′M3′M6′M9′M4M4′M5M5′M6M6′M7M7′M8M8′M9M9′M10M10′ ⟶ eliminatedWhen the initial population size isn and the number of groups is k, the number of individuals eliminated and the remainders can be assessed by (1) and (2):(1)z=n−n/kk,(2)r=n−z=n/kk,where z is the number of individuals eliminated, r is the number of remainders, n is the initial population size, and k is the number of groups.  is the rounding down symbol.After sorting and elimination, the remainders are divided into groups. The number of individuals in each group is as shown in :(3)m=r/k,where m is the number of individuals in each group.We assume that the initial population size is 100, and the number of groups is a value from 2∼10. Then, we can calculate the number of individuals eliminated and the remainders by (1) and (2), and we can calculate the number of individuals in each group by equation (3). The calculation results are shown in Table 2.Table 2
Number of individuals eliminated and remainders and the quantity of individuals in each group under different number of groups.
kzrm20100503199334010025501002064961672981484961291991110010010


### 2.2.3. Selection Operator

After sorting and grouping of individuals in the population, it is necessary to select individuals from each group. Selecting individuals with the same ranking position in each group due to the obvious divergence among the individuals in different groups after sorting and grouping. This selection operation is to avoid the randomness caused by roulette selection, and to ensure the diversity of the population to a certain extent because of the differences between individuals which are used to crossover. In the case of three groups as Table1, M1′M4′M7′ were selected at the first selection, M2′M5′M8′ were selected at the second selection, and M3′M6′M9′ were selected at the third selection. Their ranking positions in each group are 1, 2 and 3 respectively, at same intervals of individuals each selection, which can ensure that there are certain differences between individuals in each selection.


## 2.2.1. Sorting of Individuals

Initially, the objective function values of each individual are calculated. Then, all individuals in the population are sorted according to the results of the calculation. There are two cases to consider: in one case, all individuals are sorted in descending order on the basis of objective function values in the maximization problem, and in the other case, all individuals are sorted in ascending order in the minimization problem. The purpose of such sorting is to ensure that the best individual always comes first, and the worst individual comes last, regardless of whether the objective function is to find the maximum or the minimum value. The SGMOGA reduces the operating time to some extent because the algorithm directly calculates the objective function value and then sorts individuals without fitness calculation processing, which most GAs and MOGAs include.


## 2.2.2. Eliminate Worse Individuals and Divide the Remainder into Groups

In the biological world, those who are less competitive may not have the opportunity to participate in reproduction. Based on this, some worse individuals may be eliminated, and the remainder are parents to breed. After individuals sorting, the number of groups needs to be determined according to the quantity of offspring to be generated. Then, the parent individuals are distributed equally among groups. To facilitate the subsequent SGMOGA operation and ensure that each remaining individual can participate in the combination, the number of individuals in each group must be equal. This involves the determination of the initial population size and whether it is necessary to eliminate worse individuals before grouping and the quantity of individuals to be eliminated. Due to the initial population may be divided into any number of groups such as 2, 3, 4, 5 groups to reach the demands of various experiments, the initial population size should be the least common multiple (60) of the number of groups, in order to ensure that all individuals in the population can be allocated and that the number of individuals in each group is equal without worse individuals were eliminated. Assuming all individuals were divided into 2∼10 groups, the initial population size should be set to 2520 (the least common multiple of 2∼10). However, when the initial population is excessively large, with an increase in the number of groups, the population size of the offspring will be rapidly expanding, which will affect the operation speed of the GA. For the sake of reasonably controlling the operation speed of the algorithm, the initial population size should be set to an appropriate value. Then, a certain number of worse individuals are eliminated properly to ensure that the number of individuals in each group is equal in the case of any number of groups.For instance, if the population size is 10 and the number of groups is 3, all individuals in the population will be sorted according to the objective function value, and the last individual will be eliminated in order to ensure that the number of individuals each group is equal. The remaining 9 individuals will be equally allocated to 3 groups. The individuals ranking in the first 3 belong to the first group, the individuals ranking from 4th to 6th are sorted into the second group, and the last 3 individuals in the sequence are sorted into the third group. The sorting elimination and grouping process is shown in Table1.Table 1
Sorting elimination and grouping process.
Before sortingAfter sortingThe first groupThe second groupThe third groupM1M1′M1′M4′M7′M2M2′M2′M5′M8′M3M3′M3′M6′M9′M4M4′M5M5′M6M6′M7M7′M8M8′M9M9′M10M10′ ⟶ eliminatedWhen the initial population size isn and the number of groups is k, the number of individuals eliminated and the remainders can be assessed by (1) and (2):(1)z=n−n/kk,(2)r=n−z=n/kk,where z is the number of individuals eliminated, r is the number of remainders, n is the initial population size, and k is the number of groups.  is the rounding down symbol.After sorting and elimination, the remainders are divided into groups. The number of individuals in each group is as shown in :(3)m=r/k,where m is the number of individuals in each group.We assume that the initial population size is 100, and the number of groups is a value from 2∼10. Then, we can calculate the number of individuals eliminated and the remainders by (1) and (2), and we can calculate the number of individuals in each group by equation (3). The calculation results are shown in Table 2.Table 2
Number of individuals eliminated and remainders and the quantity of individuals in each group under different number of groups.
kzrm20100503199334010025501002064961672981484961291991110010010


## 2.2.3. Selection Operator

After sorting and grouping of individuals in the population, it is necessary to select individuals from each group. Selecting individuals with the same ranking position in each group due to the obvious divergence among the individuals in different groups after sorting and grouping. This selection operation is to avoid the randomness caused by roulette selection, and to ensure the diversity of the population to a certain extent because of the differences between individuals which are used to crossover. In the case of three groups as Table1, M1′M4′M7′ were selected at the first selection, M2′M5′M8′ were selected at the second selection, and M3′M6′M9′ were selected at the third selection. Their ranking positions in each group are 1, 2 and 3 respectively, at same intervals of individuals each selection, which can ensure that there are certain differences between individuals in each selection.


## 2.3. Combination Pairing Crossover Operator

### 2.3.1. Combination and Pairing

One individual from each group is chosen at a time, and they are combined to form new combinations. By makingm choices in each group with all groups having the same number of members, m combinations are obtained. The above combination operation is performed to ensure the diversity of the combination to a certain extent because of the differences among individuals in combinations. Then, individuals in new combinations can make pairing and crossover if it meets the crossover condition. We assume that the crossover condition is met, and the process of combination and pairing is shown in Table 3.Table 3
Combination and pairing process.
NumberCombinationPairing1M1′M4′M7′M1′ ↔ M4′M1′ ↔ M7′M4′ ↔ M7′2M2′M5′M8′M2′ ↔ M5′M2′ ↔ M8′M5′ ↔ M8′3M3′M6′M9′M3′ ↔ M6′M3′ ↔ M9′M6′ ↔ M9′Subsequently, the relationship between the number of new combinations, the quantity of individuals in the combination and the number of groups should be discussed. According to Tables1–3, it can be seen that the number of new combinations is equal to the number of individuals in each group, while the number of individuals in the new combination is equal to the number of groups. Thus, the number of new combinations and the quantity of individuals in each combination can be expressed by (4) and (5):(4)c=m=r/k,(5)w=k,where c is the number of new combinations, and w is the number of individuals in the new combination.In the combination, any two individuals can make a pair. Thus, the number of pairs in each combination that containsw individuals can be expressed by :(6)pairs=Cw2,where pairs is the number of pairs in each combination.If we assume that the initial population size is 100 and that the number of groups is a value from 2∼10, we can calculate the number of new combinations by (4), the number of individuals in each new combination by equation (5) and the number of pairs in each combination by equation (6). The result is shown in Table 4.Table 4
Number of individuals and pairs in each combination and the number of combinations.
kcwpairs25021333334254652051061661571472181282891193610101045


### 2.3.2. Transposons Crossover Operator

To increase the convergence speed of the global optimal solution, a transposons crossover operator was designed in this paper that emulates the gene transposition in the genome that was discovered by Nobel Laureate, Barbara McClintock, in her work on the corn plants. Transposons are also known as jumping genes, McClintock’s experimental observation indicated that there were two ways in which the transposons could jump around the genome. Two jumping ways were called cut-and-paste and copy-and-paste [44], as depicted in Figures 1 and 2, respectively. In both figures, the gray part of the chromosome represents transposons.Figure 1
Cut-and-paste transposons. (a) Cut-and-paste transposons—same chromosome (b) cut-and-paste transposons—different chromosomes.
(a)(b)Figure 2
Copy-and-paste transposons. (a) Copy-and-paste transposons—same chromosome (b) copy-and-paste transposons—different chromosomes.
(a)(b)The cut-and-paste operation is that the transposons is cut from an original position of a chromosome and pasted it into a new position. As for the copy-and-paste operation, the transposons replicates itself and the copy is pasted into a new area of the chromosome, while the original one remains unchanged [44].Based on the transposons copy-and-paste operation between different chromosomes, a transposons crossover operator was proposed in this paper. First, a condition of crossover between a pair individual was set. Paired individuals in combinations carry out the crossover operation if the random number generated was less than the crossover probabilityPc, otherwise the paired individuals were retained as offspring. When the crossover condition was met, paired individuals in combinations carry out the crossover operation and generate multi-offspring.Six offspring should be generated by the crossover or copy operation when there are three individuals in a combination. The process of the crossover or copy operation in the first combination shown in Table3 is shown in Table 5.Table 5
Process of the crossover or copy operation.
CombinationConditionOperationPairingOffspringM1′M4′M7′rand<PcCrossoverM1′,M4′child1, child2M1′,M7′child3, child4M4′,M7′child5, child6rand>PcCopyM1′ ⟶ child1, child2M4′ ⟶ child3, child4M7′ ⟶ child5, child6To compare with the MOGA, the following discussion concerns the offspring generation process by transposons crossover operator between paired individuals in a combination.According to equation (5) and (6), when the initial population was divided into k groups, each new combination contain k parents. Ck2 pairs of parents will be formed when the crossover condition was met. In order to preserve and transmit the excellent genes of excellent individuals to speed up the convergence of the algorithm, we set that the transpositions length of each pair of crossover individuals is different, that is, the transpositions length of excellent individuals in each pair of individuals is greater than that of poor individuals. Transposons crossover operator was performed each pair of individuals, firstly, the position of transposons each individual and the insertion position of the transposons in another individual should be determined. Then, the transposons of one individual was inserted into the insertion position of another individual in the paired individual. Finally, the transposons crossover operation was completed after the transposons in the paired individuals were exchanged. It was assumed that the initial population was divided into three groups, and the three parents in a combination were represented as follows:parent1 = 110011110001000parent2 = 001111001001011parent3 = 010110011010011According to the above-mentioned three individuals, they can form three pairs of parents. In the first pair of individuals composed of parent 1 and parent 2, their transposons were gene strings located between 4 and 11, 6 and 10, respectively, the insertion position were 9 and 2. child1 and child2 were generated by exchanging transposons and inserting them into the insertion position of another parent individual. The other two pairs of parent individuals in the combination also used the same transposons crossover operation to produce child3, child4, child5 and child6. The process of multi-offspring generation by the transposons crossover operator was shown in Figure3.Figure 3
Process of multi-offspring generation by transposons crossover operator.We observed that the number of multioffspring generated by SGMOGA is mainly achieved by sorting grouping selection and combination pairing crossover, which is not directly related to crossover operation times among a pair of parents. Sorting grouping selection and the combination pairing crossover operator can increase the convergence speed of the global optimal solution and maintain population diversity.


### 2.3.3. Relationship between the Number of Groups and Offspring Generated

Different from other MOGAs regarding the manner of offspring production, SGMOGA can generate offspring through introducing the sorting grouping and combination pairing mechanism into the GA, and the number of offspring produced is related to the number of groups and not related to the number of times of crossover operation among a pair of parents as in other MOGAs. Then, the relationship between the number of groups and the quantity of offspring generated will be discussed.SGMOGA can generate two offspring by crossover in one pair of parents. Therefore, the number of offspring is twice the number of parent pairs. The total number of offspring generated by the initial population can be expressed by:(7)o=2⋅pairs⋅c,where o is the total number of offspring.Because all offspring are generated byr parents, the quantitative proportional relationship between o offspring and r parents can be assessed by :(8)prop=or=2⋅pairs⋅cr=2⋅Cw2⋅r/kr=k−1,where prop is the proportion between o offspring and r parents.As previously assumed, the relationship between the number of groups and the quantity of offspring is shown in Table6.Table 6
The relationship between the number of groups and the quantity of offspring.
korprop21001001319899243001003540010046480965758898686729679792998109001009As shown in Table6, the number of offspring produced by SGMOGA was positively correlated with the number of groups. Specifically, when the number of groups is k, the number of offspring generated is k−1 times of the remaining parents after elimination. The flow chart of sorting grouping selection and the combination pairing crossover operator is shown in Figure 4.Figure 4
The flow chart of sorting grouping selection and the combination pairing crossover operator.


## 2.3.1. Combination and Pairing

One individual from each group is chosen at a time, and they are combined to form new combinations. By makingm choices in each group with all groups having the same number of members, m combinations are obtained. The above combination operation is performed to ensure the diversity of the combination to a certain extent because of the differences among individuals in combinations. Then, individuals in new combinations can make pairing and crossover if it meets the crossover condition. We assume that the crossover condition is met, and the process of combination and pairing is shown in Table 3.Table 3
Combination and pairing process.
NumberCombinationPairing1M1′M4′M7′M1′ ↔ M4′M1′ ↔ M7′M4′ ↔ M7′2M2′M5′M8′M2′ ↔ M5′M2′ ↔ M8′M5′ ↔ M8′3M3′M6′M9′M3′ ↔ M6′M3′ ↔ M9′M6′ ↔ M9′Subsequently, the relationship between the number of new combinations, the quantity of individuals in the combination and the number of groups should be discussed. According to Tables1–3, it can be seen that the number of new combinations is equal to the number of individuals in each group, while the number of individuals in the new combination is equal to the number of groups. Thus, the number of new combinations and the quantity of individuals in each combination can be expressed by (4) and (5):(4)c=m=r/k,(5)w=k,where c is the number of new combinations, and w is the number of individuals in the new combination.In the combination, any two individuals can make a pair. Thus, the number of pairs in each combination that containsw individuals can be expressed by :(6)pairs=Cw2,where pairs is the number of pairs in each combination.If we assume that the initial population size is 100 and that the number of groups is a value from 2∼10, we can calculate the number of new combinations by (4), the number of individuals in each new combination by equation (5) and the number of pairs in each combination by equation (6). The result is shown in Table 4.Table 4
Number of individuals and pairs in each combination and the number of combinations.
kcwpairs25021333334254652051061661571472181282891193610101045


## 2.3.2. Transposons Crossover Operator

To increase the convergence speed of the global optimal solution, a transposons crossover operator was designed in this paper that emulates the gene transposition in the genome that was discovered by Nobel Laureate, Barbara McClintock, in her work on the corn plants. Transposons are also known as jumping genes, McClintock’s experimental observation indicated that there were two ways in which the transposons could jump around the genome. Two jumping ways were called cut-and-paste and copy-and-paste [44], as depicted in Figures 1 and 2, respectively. In both figures, the gray part of the chromosome represents transposons.Figure 1
Cut-and-paste transposons. (a) Cut-and-paste transposons—same chromosome (b) cut-and-paste transposons—different chromosomes.
(a)(b)Figure 2
Copy-and-paste transposons. (a) Copy-and-paste transposons—same chromosome (b) copy-and-paste transposons—different chromosomes.
(a)(b)The cut-and-paste operation is that the transposons is cut from an original position of a chromosome and pasted it into a new position. As for the copy-and-paste operation, the transposons replicates itself and the copy is pasted into a new area of the chromosome, while the original one remains unchanged [44].Based on the transposons copy-and-paste operation between different chromosomes, a transposons crossover operator was proposed in this paper. First, a condition of crossover between a pair individual was set. Paired individuals in combinations carry out the crossover operation if the random number generated was less than the crossover probabilityPc, otherwise the paired individuals were retained as offspring. When the crossover condition was met, paired individuals in combinations carry out the crossover operation and generate multi-offspring.Six offspring should be generated by the crossover or copy operation when there are three individuals in a combination. The process of the crossover or copy operation in the first combination shown in Table3 is shown in Table 5.Table 5
Process of the crossover or copy operation.
CombinationConditionOperationPairingOffspringM1′M4′M7′rand<PcCrossoverM1′,M4′child1, child2M1′,M7′child3, child4M4′,M7′child5, child6rand>PcCopyM1′ ⟶ child1, child2M4′ ⟶ child3, child4M7′ ⟶ child5, child6To compare with the MOGA, the following discussion concerns the offspring generation process by transposons crossover operator between paired individuals in a combination.According to equation (5) and (6), when the initial population was divided into k groups, each new combination contain k parents. Ck2 pairs of parents will be formed when the crossover condition was met. In order to preserve and transmit the excellent genes of excellent individuals to speed up the convergence of the algorithm, we set that the transpositions length of each pair of crossover individuals is different, that is, the transpositions length of excellent individuals in each pair of individuals is greater than that of poor individuals. Transposons crossover operator was performed each pair of individuals, firstly, the position of transposons each individual and the insertion position of the transposons in another individual should be determined. Then, the transposons of one individual was inserted into the insertion position of another individual in the paired individual. Finally, the transposons crossover operation was completed after the transposons in the paired individuals were exchanged. It was assumed that the initial population was divided into three groups, and the three parents in a combination were represented as follows:parent1 = 110011110001000parent2 = 001111001001011parent3 = 010110011010011According to the above-mentioned three individuals, they can form three pairs of parents. In the first pair of individuals composed of parent 1 and parent 2, their transposons were gene strings located between 4 and 11, 6 and 10, respectively, the insertion position were 9 and 2. child1 and child2 were generated by exchanging transposons and inserting them into the insertion position of another parent individual. The other two pairs of parent individuals in the combination also used the same transposons crossover operation to produce child3, child4, child5 and child6. The process of multi-offspring generation by the transposons crossover operator was shown in Figure3.Figure 3
Process of multi-offspring generation by transposons crossover operator.We observed that the number of multioffspring generated by SGMOGA is mainly achieved by sorting grouping selection and combination pairing crossover, which is not directly related to crossover operation times among a pair of parents. Sorting grouping selection and the combination pairing crossover operator can increase the convergence speed of the global optimal solution and maintain population diversity.


## 2.3.3. Relationship between the Number of Groups and Offspring Generated

Different from other MOGAs regarding the manner of offspring production, SGMOGA can generate offspring through introducing the sorting grouping and combination pairing mechanism into the GA, and the number of offspring produced is related to the number of groups and not related to the number of times of crossover operation among a pair of parents as in other MOGAs. Then, the relationship between the number of groups and the quantity of offspring generated will be discussed.SGMOGA can generate two offspring by crossover in one pair of parents. Therefore, the number of offspring is twice the number of parent pairs. The total number of offspring generated by the initial population can be expressed by:(7)o=2⋅pairs⋅c,where o is the total number of offspring.Because all offspring are generated byr parents, the quantitative proportional relationship between o offspring and r parents can be assessed by :(8)prop=or=2⋅pairs⋅cr=2⋅Cw2⋅r/kr=k−1,where prop is the proportion between o offspring and r parents.As previously assumed, the relationship between the number of groups and the quantity of offspring is shown in Table6.Table 6
The relationship between the number of groups and the quantity of offspring.
korprop21001001319899243001003540010046480965758898686729679792998109001009As shown in Table6, the number of offspring produced by SGMOGA was positively correlated with the number of groups. Specifically, when the number of groups is k, the number of offspring generated is k−1 times of the remaining parents after elimination. The flow chart of sorting grouping selection and the combination pairing crossover operator is shown in Figure 4.Figure 4
The flow chart of sorting grouping selection and the combination pairing crossover operator.


## 2.4. Mutation Operator

The SGMOGA approach proposed in this paper adopt the basic bit mutation operator, that is, a gene bit randomly designated on the binary string is modified with the mutation operator when the generated random number is less than the mutation probability value. The basic bit mutation operator can realize the local search function of the algorithm, which is a common method of a GA under binary coding.


## 2.5. Population Competition

For SGMOGA, it is necessary to keep the population size unchanged; otherwise, the running time of the algorithm will increase significantly with the increase of the number of offspring. Therefore, population competition is introduced into SGMOGA for the sake of maintaining an unchanged parent population size; therefore, the excellent individuals within the population can survive better, which will promote the population to develop in a better direction. In this process,z worse individuals are first eliminated after sorting n individuals according their objective function value; then r individuals are divided into k groups. There will be r parents participating in the combination pairing crossover operation. As previously analyzed, the number of offspring generated after combination pairing crossover is k−1 times as many as r parents, so the number of offspring generated after crossover by SGMOGA is illustrated by :(9)n1=k−1r,where n1 is the number of offspring generated by the crossover of r parents.(2) is substituted into equation (9) and it can be changed into :(10)n1=k−1n/kk.For the sake of population competition,s elitists are picked from the parent population and put into the offspring population. After crossover and the first-time elitist reserved, the offspring population size is shown in :(11)n2=s+n1=s+k−1n/kk,where n2 is the offspring population size, s is the number of the first-time elitist reserved.To avoid the damage of mutation to excellent individuals,s elitists are first picked from n2 individuals, then the mutation operation is performed on n1 individuals generated by crossover. Finally, after mutation and the second-time elitist reserved, there are n2 individuals generated.It is thus clear from (11) and Table 6 that the population size of offspring in SGMOGA is far larger than the initial population size. This indicates a larger size disparity between the size of parents and offspring with an increasing number of groups. In this large population, not only many individuals with poor survivability will exist, but also the operation speed of the algorithm will be seriously affected. Therefore, we need to eliminate some worse individuals through population competition. Consequently, after mutation and the second-time elitist reserved, picking n optimal individuals from n2 offspring, will form a new parent population such that the population size is unchangeable at each iteration, which can speed up the algorithm computation.


## 2.6. Catastrophe Mechanism

Catastrophe is the extinction of a large number of species or individuals due to major catastrophic accidents in biological evolution. Only species or individuals with strong adaptability can survive and reproduce [45]. In order to avoid falling into local convergence, catastrophe mechanism is introduced into this algorithm to make the population restart diversified search, so as to find the global optimal solution. In SGMOGA, when the optimal value of the objective function for three consecutive generations does not change, the catastrophe mechanism start work, which the excellent individual is retained, while some worse are reinitialized when the generated random number is less than the catastrophe probability, to maintain the diversity of the population and restart diversified search.


## 3. Evolutionary Strategy of SGMOGA

The parameters of SGMOGA are initiated such that the initial population size is equal ton, the probability of crossover, mutation and catastrophe are pc, pm and pca, respectively. The SGMOGA will run N times.The succeeding evolutionary strategy of the SGMOGA proposed in this paper is as follows:(1)
Initial populations having fixed order are randomly generated as parents; each population has the same size, which is equal ton.(2)
The sorting operation is performed, that is, all individuals in the population are sorted in order (descending order in the maximization problem or ascending order in the minimization problem) according to their objective function values. Go to the next step if the stop condition is not met. Otherwise, output the best objective function value and terminate the loop.(3)
Selects elitist members with the best ranking among n parents.(4)
The catastrophe mechanism start work and the new generated population was reordered, only when the optimal value of the objective function for three consecutive generations does not change.(5)
The grouping operation is carried out after eliminatingz worse members, and remaining parent individuals are distributed equally into k groups.(6)
Select one individual from each group at a time, choosingm times in total.(7)
Combiningk individuals that are selected from k groups at a time. Choosing m times for each group in total, so m combinations are obtained.(8)
Pairing operations are performed after the combination operation. Any two individuals in a new combination are paired off if the crossover condition is met.(9)
For the crossover operation, use the transposons crossover operator in each pair of individuals if the crossover condition is met; otherwise, copy individual in the combination. Finally, generaten1 offspring in total.(10)
Generate a new population consisting of then1 offspring generated in step (9) and the s elitist members picked in Step (3).(11)
Selects elitist members with the best ranking among n1+s members in the new population generated in Step (10).(12)
Using the mutation operator, modify then1 offspring generated by the crossover operation in Step (9).(13)
Regenerate the new population consisting of then1 mutated offspring in Step (12) and s elitist members chosen in Step (11).(14)
Finally, selectn members with the best ranking within the last population as the new parent population. Go to Step (2) to start the next loop.The evolutionary strategy flow diagram for SGMOGA is shown in Figure5.Figure 5
The evolutionary strategy flow diagram for SGMOGA.


## 4. Algorithmic Testing and Analysis

To verify the validity and feasibility of SGMOGA presented in this paper, it is compared with simple genetic algorithms (SGA) [46] and the existing MOGA. SGA uses binary encoding, roulette wheel selection, single-point crossover and the basic bit mutation operator [46]. Among the existing references of MOGA, literature [39, 40] adopted the single-point crossover operator and two-point crossover operator to generate multi-offspring, respectively, and the two studies have made a great amount of comparative studies of MOGA to verify the effectiveness and feasibility of the algorithm. Therefore, SGMOGA will be compared with SGA, and with the two MOGAs proposed respectively in [39, 40]. In the following, MOGA which the adopts single-point crossover operator in [39] is abbreviated as S-MOGA, and MOGA which use the two-point crossover operator in [40] is abbreviated as T-MOGA.


### 4.1. Parameter Initialization

To ensure the fairness of the algorithms in the performance comparison process, the relevant parameter initialization as shown in Table7.Table 7
Parameter initialization.
ParametersSGAS-MOGAT-MOGASGMOGAn100100100100pc0.90.90.90.9pm0.10.10.10.1pca———0.1s—101010ε10−410−410−410−4prop—222∗n is initial population size; pc, pm and pca are the probability of crossover, mutation and catastrophe respectively; s is the number of elitists reserved; ε is iteration termination condition; prop is the population size proportion of parents to offspring.


### 4.2. Iteration Termination Condition

The iteration termination condition for the above four algorithms can be defined by:(12)fi−fi∗≤εi,i=1,2,…,p,where fi is the optimal value of the ith test function obtained by four algorithms, fi∗ is the globally optimal value of the ith test function in theory, and εi is the precision for the ith test function.


### 4.3. Test Functions

To verify the validity and feasibility of SGMOGA, 29 benchmark test functions included in CEC2017 [47] were selected to test the effectiveness and applicability of the improved algorithm (function f2 has poor stability and was excluded by CEC2017 [47], which will not be tested in this paper), these functions’ search range is [−100, 100]D. They were shown in Table 8.Table 8
Test functions.
No.FunctionsOptimal valueNo.FunctionsOptimal valuef1Shifted and rotated bent cigar function100f17Hybrid function 6 (N = 5)1700f3Shifted and rotated zakharov function300f18Hybrid function 6 (N = 5)1800f4Shifted and rotated Rosenbrock’s function400f19Hybrid function 6 (N = 5)1900f5Shifted and rotated Rastrigin’s function500f20Hybrid function 6 (N = 6)2000f6Shifted and rotated expanded Scaffer’s F6 function600f21Composition function 1 (N = 3)2100f7Shifted and rotated lunacek Bi_Rastrigin function700f22Composition function 2 (N = 3)2200f8Shifted and rotated non-continuous Rastrigin’s function800f23Composition function 3 (N = 4)2300f9Shifted and rotated levy function900f24Composition function 4 (N = 4)2400f10Shifted and rotated Schwefel’s function1000f25Composition function 5 (N = 5)2500f11Hybrid function 1 (N = 3)1100f26Composition function 6 (N = 5)2600f12Hybrid function 2 (N = 3)1200f27Composition function 7 (N = 6)2700f13Hybrid function 3 (N = 3)1300f28Composition function 8 (N = 6)2800f14Hybrid function 4 (N = 4)1400f29Composition function 9 (N = 3)2900f15Hybrid function 5 (N = 4)1500f30Composition function 10 (N = 3)3000f16Hybrid function 6 (N = 4)1600


### 4.4. Test Results

#### 4.4.1. Verification of Operator Basic Functions

In order to study the role of each operator in SGMOGA, we disabled each operator in turn, and other operators were used normally. The algorithm that disabling sorting grouping selection operator but using roulette selection was marked as Test1, and disabling the transposons crossover operator, mutation operator, and catastrophe mechanism were recorded as Test2, Test3, and Test4, respectively. SGMOGA that using all operators at the same time was denoted by Test5. The experiment selectedf11∼f20 from the CEC2017 for testing. The evaluation indicators were the average population diversity (div) [48] and the average value (mean) of the optimal value. The results were shown in Tables 9 and 10.Table 9
Experimental results of the evaluation indicator (div).
DivTest1Test2Test3Test4Test5f115.26E + 031.14E + 049.07E + 028.82E + 039.04E + 03f124.18E + 031.13E + 048.46E + 027.35E + 037.74E + 03f135.19E + 031.16E + 041.66E + 038.38E + 038.65E + 03f145.63E + 031.17E + 042.16E + 038.95E + 039.23E + 03f155.56E + 031.17E + 041.66E + 039.04E + 039.26E + 03f164.88E + 031.16E + 041.10E + 038.10E + 038.34E + 03f175.19E + 031.15E + 041.33E + 038.44E + 038.65E + 03f184.31E + 038.92E + 031.60E + 038.06E + 038.30E + 03f194.84E + 031.16E + 042.00E + 038.77E + 038.92E + 03f205.71E + 038.46E + 031.58E + 039.09E + 039.35E + 03Table 10
Experimental results of the evaluation indicator (mean).
MeanTest1Test2Test3Test4Test5f111.12E + 031.16E + 031.15E + 031.11E + 031.11E + 03f122.99E + 052.70E + 063.01E + 053.22E + 052.90E + 05f136.66E + 038.66E + 036.77E + 035.33E + 035.19E + 03f141.47E + 031.48E + 031.48E + 031.43E + 031.43E + 03f151.96E + 031.81E + 032.24E + 031.66E + 031.66E + 03f161.63E + 031.64E + 031.68E + 031.61E + 031.61E + 03f171.72E + 031.75E + 031.74E + 031.72E + 031.72E + 03f186.62E + 031.18E + 048.50E + 038.50E + 038.14E + 03f192.05E + 032.05E + 032.58E + 031.93E + 031.93E + 03f202.02E + 032.04E + 032.04E + 032.02E + 032.02E + 03It can be seen from the Table9 that the diversity of Test2 for most functions other than f20 was the best among all the experiments, which shown that the simultaneous use of sorting grouping selection operator, mutation operator and catastrophe mechanism can improve population diversity. Knowing from the comparison results of Test1, Test3 and Test4 with Test5 respectively, in improving population diversity, mutation operator was better than sorting grouping selection operator, and sorting grouping selection operator was superior to catastrophe mechanism. From Table 10, we can see that Test5 has obtained the best experimental results, which proved that SGMOGA using all operators at the same time has the best performance and transposons crossover operator has strong search capability.


#### 4.4.2. The Average Optimal Value, Standard Deviation and the Number of Times When Converging to the Global Optimal Solution

To obtain a fair performance comparison, 29 test functions of CEC2017 were used as examples and each test function was carried out 50 times with MATLAB R2016a. The experimental evaluation indicators were the average optimal value, minimum, maximum, standard deviation, and the number of times when converging to the global optimal solution under certain iterations, which can be represented respectively by avg, min, max, std and nos. The experimental results off1∼f10 for 2D problems are shown in Table 11. The experimental results of f11∼f20 for 10D problems and f21∼f30 for 30D problems at 10000 iterations are shown in Table 12.Table 11
The experimental results off1∼f10 for 2D problems.
f/genAlgorithmNosAvgMaxStdf1/10000SGA21.4273E + 022.5838E + 026.4120E + 01S-MOGA31.4515E + 022.5797E + 026.3623E + 01T-MOGA11.4816E + 022.5836E + 026.6638E + 01SGMOGA351.0000E + 021.0009E + 021.6911E − 02f3/100SGA143.1565E + 023.6039E + 022.1979E + 01S-MOGA203.1239E + 023.5893Ev022.0929E + 01T-MOGA213.1239E + 023.5891Ev022.0924E + 01SGMOGA503.0000E + 023.0000E + 022.4198E − 06f4/50SGA54.0002E + 024.0014E + 023.3144E − 02S-MOGA114.0001E + 024.0006E + 022.0530E − 02T-MOGA134.0001E + 024.0006E + 022.0391E − 02SGMOGA454.0000E + 024.0001E + 028.0654E − 04f5/100SGA105.0024E + 025.0100E + 024.1927E − 01S-MOGA195.0016E + 025.0116E + 023.7238E − 01T-MOGA205.0014E + 025.0145E + 023.7022E − 01SGMOGA415.0000E + 025.0000E + 021.0817E − 03f6/500SGA156.0023E + 026.0053E + 022.5440E − 01S-MOGA186.0021E + 026.0002E + 022.4970E − 01T-MOGA166.0019E + 026.0053E + 022.4387E − 01SGMOGA486.0000E + 026.0000E + 024.4401E − 05f7/500SGA37.0183E + 027.0304E + 028.6301E − 01S-MOGA27.0160E + 027.0263E + 028.5167E − 01T-MOGA37.0155E + 027.0298E + 029.2149E − 01SGMOGA507.0000E + 027.0000E + 026.2974E − 09f8/500SGA148.0065E + 028.0398E + 028.8120E − 01S-MOGA20 8.0042E + 028.0199E + 025.1377E − 01T-MOGA178.0048E + 028.0398E + 027.2909E − 01SGMOGA478.0000E + 028.0000E + 021.3008E − 04f9/100SGA129.0020E + 029.0095E + 022.7031E − 01S-MOGA109.0018E + 029.0065E + 022.5197E − 01T-MOGA149.0012E + 029.0065E + 022.1634E − 01SGMOGA359.0000E + 029.0010E + 021.3517E − 02f10/1000SGA21.0067E + 031.0171E + 038.0015E + 00S-MOGA31.0054E + 031.0171E + 037.5548E + 00T-MOGA61.0049E + 031.0171E + 037.4838E + 00SGMOGA281.0000E + 031.0002E + 033.4186E − 02Table 12
The experimental results off11∼f20 for 10D problems and f21∼f30 for 30D problems.
fAlgorithmAvgMinMaxStdf11SGA1.1294E + 031.1074E + 031291.0741675.3684E + 01S-MOGA1.1379E + 031.1043E + 031296.2561086.4838E + 01T-MOGA1.1092E + 031.1018E + 031119.5695935.0203E + 00SGMOGA1.1078E + 031.1021E + 031116.7007463.9382E + 00f12SGA1.4350E + 057.1900E + 039.4500E + 052.4850E + 05S-MOGA1.2578E + 057.9300E + 031.2000E + 062.6381E + 05T-MOGA2.2080E + 052.6400E + 031.8500E + 065.2809E + 05SGMOGA1.9592E + 042.1400E + 036.5000E + 041.8463E + 04f13SGA1.0082E + 041.3300E + 034.5200E + 041.0974E + 04S-MOGA7.7692E + 031.3200E + 032.0400E + 045.3334E + 03T-MOGA4.7878E + 031.3200E + 031.3700E + 043.9643E + 03SGMOGA1.5631E + 031.3200E + 032.1600E + 032.1820E + 02f14SGA2.0156E + 031.4084E + 035.0965E + 031.2556E + 03S-MOGA1.5349E + 031.4028E + 032.4366E + 032.2322E + 02T-MOGA1.4824E + 031.4039E + 031.6566E + 038.0107E + 01SGMOGA1.4130E + 031.4030E + 031.4387E + 038.3338E + 00f15SGA1.8449E + 031.5066E + 034.2233E + 036.0847E + 02S-MOGA1.7632E + 031.5034E + 034.2166E + 035.9500E + 02T-MOGA1.9437E + 031.5017E + 034.9532E + 039.0751E + 02SGMOGA1.5868E + 031.5019E + 031.9327E + 031.4655E + 02f16SGA1.6189E + 031.6005E + 031.7465E + 033.9286E + 01S-MOGA1.6235E + 031.6010E + 031.7477E + 034.7449E + 01T-MOGA1.6190E + 031.6004E + 031.7327E + 033.6837E + 01SGMOGA1.6016E + 031.6001E + 031.6129E + 032.6508E + 00f17SGA1.7097E + 031.7025E + 031.7342E + 037.6515E + 00S-MOGA1.7096E + 031.7017E + 031.7234E + 035.9522E + 00T-MOGA1.7093E + 031.7021E + 031.7369E + 038.8750E + 00SGMOGA1.7079E + 031.7012E + 031.7189E + 036.8870E + 00f18SGA4.4026E + 031.8812E + 031.2589E + 042.5799E + 03S-MOGA4.0847E + 031.8507E + 031.2810E + 042.5931E + 03T-MOGA4.8871E + 031.8531E + 031.2513E + 042.9346E + 03SGMOGA2.9571E + 031.8126E + 038.9009E + 031.9532E + 03f19SGA1.9159E + 031.9025E + 031.9446E + 031.5818E + 01S-MOGA1.9208E + 031.9017E + 031.9497E + 031.8795E + 01T-MOGA1.9203E + 031.9017E + 031.9635E + 032.0544E + 01SGMOGA1.9076E + 031.9016E + 031.9433E + 038.7784E + 00f20SGA2.0088E + 032.0013E + 032.0228E + 036.9580E + 00S-MOGA2.0095E + 032.0012E + 032.0244E + 036.6974E + 00T-MOGA2.0077E + 032.0012E + 032.0198E + 035.6305E + 00SGMOGA2.0037E + 032.0006E + 032.0204E + 034.1985E + 00f21SGA2.4831E + 032.4513E + 032.5066E + 031.2831E + 01S-MOGA2.4755E + 032.4384E + 032.4919E + 031.3226E + 01T-MOGA2.4787E + 032.4530E + 032.5062E + 031.4300E + 01SGMOGA2.3935E + 032.3598E + 032.4268E + 031.7609E + 01f22SGA4.2946E + 032.6184E + 038.3313E + 032.4313E + 03S-MOGA3.4246E + 032.5458E + 037.7281E + 031.7958E + 03T-MOGA2.8843E + 032.5425E + 037.4651E + 031.0524E + 03SGMOGA2.5643E + 032.4235E + 032.7233E + 039.5558E + 01f23SGA2.8492E + 032.8286E + 032.8644E + 031.1295E + 01S-MOGA2.8319E + 032.8069E + 032.8582E + 031.3322E + 01T-MOGA2.8304E + 032.8026E + 032.8600E + 031.7635E + 01SGMOGA2.7506E + 032.7141E + 032.7933E + 032.0916E + 01f24SGA3.0336E + 033.0096E + 033.0610E + 031.2011E + 01S-MOGA3.0242E + 033.0085E + 033.0558E + 031.1581E + 01T-MOGA3.0246E + 033.0399E + 033.0446E + 031.3840E + 01SGMOGA2.9326E + 032.8895E + 032.9995E + 033.0003E + 01f25SGA3.0475E + 032.9993E + 033.1783E + 033.9304E + 01S-MOGA3.0352E + 032.9797E + 033.1351E + 034.1771E + 01T-MOGA3.0367E + 032.9838E + 033.1921E + 035.3587E + 01SGMOGA2.9830E + 032.9366E + 033.0328E + 033.1622E + 01f26SGA5.7855E + 035.5328E + 036.0841E + 031.3774E + 02S-MOGA5.6551E + 035.4031E + 035.8844E + 031.5856E + 02T-MOGA5.4773E + 033.6355E + 035.9961E + 034.6909E + 02SGMOGA4.7061E + 033.5117E + 035.3600E + 034.4765E + 02f27SGA3.2528E + 033.2433E + 033.2592E + 034.5962E + 00S-MOGA3.2448E + 033.2101E + 033.2633E + 031.3109E + 01T-MOGA3.2427E + 033.2210E + 033.2607E + 031.0229E + 01SGMOGA3.2341E + 033.2089E + 033.2623E + 031.5596E + 01f28SGA3.4106E + 033.3169E + 033.4993E + 034.7750E + 01S-MOGA3.3893E + 033.3368E + 033.5259E + 034.6290E + 01T-MOGA3.3802E + 033.3167E + 033.4629E + 033.0698E + 01SGMOGA3.3963E + 033.2987E + 033.5851E + 038.6236E + 01f29SGA3.7879E + 033.5729E + 034.0574E + 031.4114E + 02S-MOGA3.6960E + 033.5055E + 033.9562E + 031.4737E + 02T-MOGA3.7258E + 033.5218E + 033.9474E + 031.3105E + 02SGMOGA3.6926E + 033.3526E + 033.9317E + 031.6212E + 02f30SGA2.7420E + 061.1300E + 052.2400E + 075.6133E + 06S-MOGA2.1569E + 068.7500E + 041.1700E + 073.1870E + 06T-MOGA8.8745E + 059.4700E + 046.0200E + 061.4033E + 06SGMOGA1.7635E + 052.4500E + 046.0500E + 051.3356E + 05As shown in Table11, the number of times that SGMOGA has achieved the global optimal solution was obviously more than that achieved by SGA, S-MOGA and T-MOGA under the same iterations. After the same amount of iterations, all four algorithms have reached the theoretically optimal solution at least once, so that no list the indicator (min) in Table 11. SGMOGA not only outperformed other algorithms on the other three indicators (avg), (max) and (std), but also the average optimal value accuracy of SGMOGA was less than 10–2 that other algorithms have not reached.It can be seen from Table12 that the average optimal value of 19 functions except f28 in SGMOGA were better than other algorithms. In indicator (min), SGMOGA was superior to other algorithms in 17 functions apart from f11, f14 and f15. The indicator (max) of SGMOGA on 17 functions except f20, f27 and f28 have surpass other algorithms. About standard deviation, SGMOGA to excel others on 12 functions.To better analyze the performance of the algorithm, we used boxplot to represent the statistical results of each algorithm for test problem by Figure6. Boxplot is an important tool for statistical analysis in the field of economics. It can well reflect the statistical distribution of data. As depicted in Figure 6, the results of SGMOGA on test functions were more concentrated and close to the optimal than the comparison algorithm, which shown that SGMOGA has better search performance and robustness.Figure 6
Boxplots of the results obtained by four algorithms. (a)f11 (b) f12 (c) f13 (d) f14 (e) f15 (f) f16 (g) f17 (h) f18 (i) f19 (j) f20 (k) f21 (l) f22.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)


#### 4.4.3. Significance Detection

T-test (at significance level of 0.05) was carried out by using the average optimal value and standard deviation to verify whether there is significant difference between improved algorithm and comparison algorithm. The T-test results are shown in Table 13. Symbol “s” in the Table 13 represent the result of T-test. Specifically, “+” indicates that the performance of SGMOGA is significantly better than the comparison algorithm; “−” expresses SGMOA is significantly worse than comparison algorithm; “=” means that the two algorithms have the same performance. B/W/E respectively indicates the number of functions that SGMOGA has significantly better, worse and equivalent than the comparison algorithm.Table 13
The results ofT-test.
fSGAS-MOGAT-MOGAtpstpstpsf1−4.71220.0000+−5.01800.0000+−5.11030.0000+f3−5.03490.0000+−4.18610.0000+−4.18710.0000+f4−4.26560.0000+−3.44160.0004+−3.46500.0004+f5−4.04760.0001+−3.03820.0015+−2.67390.0044+f6−6.39290.0000+−5.94680.0000+−5.50910.0000+f7−14.99410.0000+−13.28410.0000+−11.89390.0000+f8−5.21580.0000+−5.78050.0000+−4.65530.0000+f9−5.22530.0000+−5.04410.0000+−3.91460.0001+f10−5.92090.0000+−5.05420.0000+−4.62970.0000+f11−2.83750.0028+−3.27660.0007+−1.55150.0620=f12−3.51610.0003+−2.83930.0027+−2.69250.0042+f13−5.48800.0000+−8.22120.0000+−5.74320.0000+f14−3.39350.0005+−3.85880.0001+−6.09310.0000+f15−2.91600.0022+−2.03550.0222+−2.74530.0036+f16−3.10680.0012+−3.25860.0008+−3.33140.0006+f17−1.23640.1096=−1.32060.0949=−0.88120.1902=f18−3.15870.0011+−2.45600.0079+−3.87130.0001+f19−3.24420.0008+−4.49950.0000+−4.01960.0001+f20−4.43760.0000+−5.18840.0000+−4.02710.0001+f21−29.07890.0000+−26.32850.0000+−26.55850.0000+f22−5.02840.0000+−3.38270.0005+−2.14130.0174+f23−29.33030.0000+−23.18220.0000+−20.62530.0000+f24−22.09850.0000+−20.13990.0000+−19.68870.0000+f25−9.04110.0000+−7.04540.0000+−6.10270.0000+f26−16.29620.0000+−14.13020.0000+−8.41010.0000+f27−8.13260.0000+−3.71370.0002+−3.26040.0008+f28−1.02580.1538=0.50570.3071=1.24370.1083=f29−3.13500.0011+−0.10970.4564=−1.12610.1314=f30−3.23100.0008+−4.39040.0000+−3.56700.0003+B/W/E27/0/226/0/325/0/4According to theT-test results in Table 13, at the significance level 0.05, SGMOGA was significantly better than SGA in 27 functions, remarkably superior to S-MOGA in 26 functions and markedly better than T-MOGA in 25 functions. And it can be seen that the performance of SGMOGA was almost the same as that of the comparison algorithm in 2, 3 and 4 functions, respectively. There was not functions significantly weaker than the comparison algorithms. Compared with SGA, S-MOGA and T-MOGA, SGMOGA shown better comprehensive performance and it was more stable.


#### 4.4.4. The Average Optimal Value under Different Iterations

To further illustrate that SGMOGA proposed in this paper improved the search performance of genetic algorithms, the seven test functions were operated respectively, and the average optimal value under different iterations was obtained to verify the search efficiency of the improved algorithm. In order to compare the convergence of four algorithms more intuitively, Figures7–13 described the convergence curves of the four algorithms on seven test functions. As shown in Figures 7–13, it was obvious that the convergence speed of SGMOGA was significantly faster than the comparison algorithm and that SGMOGA proposed in this paper improved the search performance compared with SGA, S-MOGA and T-MOGA.Figure 7
The average optimal value of functionf1.Figure 8
The average optimal value of functionf3.Figure 9
The average optimal value of functionf4.Figure 10
The average optimal value of functionf6.Figure 11
The average optimal value of functionf7.Figure 12
The average optimal value of functionf8.Figure 13
The average optimal value of functionf10.


#### 4.4.5. The Average Running Time of Iterations

To verify the convergence rate of the algorithm, the average running time of iterations when converging to the optimal solution were calculated as follows: when obtaining the calculation an accuracy of10−4, the running time of iterations at the ith calculation is t(i), i = 1, 2, …, N, where N is the number of runs. Then, the average running time of iterations can be calculated by :(13)tav=1N∑1Nti,where tav is the average running time of iterations.The average running time of iterations converging to the optimal solution were shown in Table14.Table 14
Average running time of iterations.
fSGAS-MOGAT-MOGASGMOGAf183.2110128.9689106.135817.8289f39.48228.27097.21730.1726f437.573536.226128.85670.1297f536.696037.399241.92720.1971f625.150651.265432.205210.6908f752.702947.575445.08730.3680f83.14811.51982.41270.1230f951.182336.930049.09240.2491f1092.3668108.6970128.41598.4093According to the experimental results in Table14, SGMOA reduced the average running time by 78.57%, 98.18%, 99.65%, 99.46%, 57.49%, 99.30%, 96.09%, 99.51% and 90.90% in comparison to SGA. Compared with S-MOGA, the average running time decrease rate were 86.18%, 97.91%, 99.64%, 99.47%, 79.15%, 99.23%, 91.91%, 99.33% and 92.26%. Moreover, this indicator was down by 83.20%, 97.61%, 99.55%, 99.53%, 66.80%, 99.18%, 94.90%, 99.49% and 93.45% relative to T-MOGA. In terms of the average running time when converging to the optimal solution, compared with SGA, S-MOGA and T-MOGA, the decline range of SGMOGA for f1∼f10 was 57.49%∼99.65%, 79.15%∼99.64% and 66.80%∼99.55% respectively. And this made the point that the average running time of SGMOGA was significantly superior to SGA, S-MOGA and T-MOGA. The results shown that SGMOGA has faster convergence to the global optimal solution.


#### 4.4.6. Performance of the Proposed Method with Different Parameters

To verify the performance of the proposed method under different population sizes, and crossover and mutation values of the GA, experiments were conducted with one parameter changed and the others parameters unchanged. The values of unchanged parameters were the same as that of parameters for initialization above. The parameter values of the experiment were as follows: (1) population sizen: 50, 100, 150, 200, 300; (2) crossover value pc: 0.6, 0.7, 0.8, 0.9, 1; and (3) mutation value pm: 0.01, 0.05, 0.1, 0.15, 0.2.Takingf3 as an example, the average running time of iterations converging to the optimal solution under different population sizes, crossover values and mutation values were shown in Tables 15–17, respectively.Table 15
Average running time under different population sizes.
nSGAS-MOGAT-MOGASGMOGA509.72886.61955.66140.18151003.79883.30684.91600.17261505.62503.23452.52990.30682002.57773.01583.43420.36863003.43143.77993.80530.4774Table 16
Average running time under different crossover values.
pcSGAS-MOGAT-MOGASGMOGA0.66.80856.17275.36450.20780.75.52485.77244.37290.20320.85.99305.23464.29360.19360.93.79883.30684.91600.17261.07.14948.44085.24620.2479Table 17
Average running time under different mutation values.
pmSGAS-MOGAT-MOGASGMOGA0.0141.997147.373356.14100.28350.0531.501724.483330.11210.20380.103.79883.30684.91600.17260.151.00940.69630.90940.24260.201.03400.85111.04270.2547Thus, it was verified that the performance of SGMOGA with different population sizes and crossover and mutation values was superior to the performances of SGA, S-MOGA and T-MOGA. From the effects of GA parameters on the performance of the proposed method, SGMOGA was shown to have the best performance with population size equal to 100, crossover value equal to 0.9 and mutation value equal to 0.1.As the above experimental series and analysis showed, the proposed SGMOGA algorithm is an effective algorithm for function optimization. The reasons are concluded as follows. First, sorting grouping selection and combination paring operations are used before crossover, which easily implemented and enhanced the diversity of the population. Second, different from traditional MOGA, the mulit-offspring generated by transposons crossover operator in different pairs of parents instead of repeat crossover in same pair of parents, to avoid inbreeding and premature convergence. In addition, the use of transposons crossover operator would be conducive to transfer excellent genes to the next generation and accelerate convergence. Third, at the same time of local search using mutation operator, to avoid falling into local convergence, catastrophe mechanism is introduced into this algorithm to make the population restart diversified search, to find the global optimal solution. Finally, the strategy of elitism and population competition are used at the same time, to protect excellent individuals and enhance the guidance of the algorithm. Moreover, it can avoid the expansion of population size to affect the operation efficiency of SGMOGA algorithm. Therefore, the SGMOGA algorithm, which is supposed to take advantage of previous knowledge of MOGA, the theory of transposons and catastrophe, the strategy of elitism and population competition to create a new variant, is an effective GA algorithm for function optimization.


## 4.1. Parameter Initialization

To ensure the fairness of the algorithms in the performance comparison process, the relevant parameter initialization as shown in Table7.Table 7
Parameter initialization.
ParametersSGAS-MOGAT-MOGASGMOGAn100100100100pc0.90.90.90.9pm0.10.10.10.1pca———0.1s—101010ε10−410−410−410−4prop—222∗n is initial population size; pc, pm and pca are the probability of crossover, mutation and catastrophe respectively; s is the number of elitists reserved; ε is iteration termination condition; prop is the population size proportion of parents to offspring.


## 4.2. Iteration Termination Condition

The iteration termination condition for the above four algorithms can be defined by:(12)fi−fi∗≤εi,i=1,2,…,p,where fi is the optimal value of the ith test function obtained by four algorithms, fi∗ is the globally optimal value of the ith test function in theory, and εi is the precision for the ith test function.


## 4.3. Test Functions

To verify the validity and feasibility of SGMOGA, 29 benchmark test functions included in CEC2017 [47] were selected to test the effectiveness and applicability of the improved algorithm (function f2 has poor stability and was excluded by CEC2017 [47], which will not be tested in this paper), these functions’ search range is [−100, 100]D. They were shown in Table 8.Table 8
Test functions.
No.FunctionsOptimal valueNo.FunctionsOptimal valuef1Shifted and rotated bent cigar function100f17Hybrid function 6 (N = 5)1700f3Shifted and rotated zakharov function300f18Hybrid function 6 (N = 5)1800f4Shifted and rotated Rosenbrock’s function400f19Hybrid function 6 (N = 5)1900f5Shifted and rotated Rastrigin’s function500f20Hybrid function 6 (N = 6)2000f6Shifted and rotated expanded Scaffer’s F6 function600f21Composition function 1 (N = 3)2100f7Shifted and rotated lunacek Bi_Rastrigin function700f22Composition function 2 (N = 3)2200f8Shifted and rotated non-continuous Rastrigin’s function800f23Composition function 3 (N = 4)2300f9Shifted and rotated levy function900f24Composition function 4 (N = 4)2400f10Shifted and rotated Schwefel’s function1000f25Composition function 5 (N = 5)2500f11Hybrid function 1 (N = 3)1100f26Composition function 6 (N = 5)2600f12Hybrid function 2 (N = 3)1200f27Composition function 7 (N = 6)2700f13Hybrid function 3 (N = 3)1300f28Composition function 8 (N = 6)2800f14Hybrid function 4 (N = 4)1400f29Composition function 9 (N = 3)2900f15Hybrid function 5 (N = 4)1500f30Composition function 10 (N = 3)3000f16Hybrid function 6 (N = 4)1600


## 4.4. Test Results

### 4.4.1. Verification of Operator Basic Functions

In order to study the role of each operator in SGMOGA, we disabled each operator in turn, and other operators were used normally. The algorithm that disabling sorting grouping selection operator but using roulette selection was marked as Test1, and disabling the transposons crossover operator, mutation operator, and catastrophe mechanism were recorded as Test2, Test3, and Test4, respectively. SGMOGA that using all operators at the same time was denoted by Test5. The experiment selectedf11∼f20 from the CEC2017 for testing. The evaluation indicators were the average population diversity (div) [48] and the average value (mean) of the optimal value. The results were shown in Tables 9 and 10.Table 9
Experimental results of the evaluation indicator (div).
DivTest1Test2Test3Test4Test5f115.26E + 031.14E + 049.07E + 028.82E + 039.04E + 03f124.18E + 031.13E + 048.46E + 027.35E + 037.74E + 03f135.19E + 031.16E + 041.66E + 038.38E + 038.65E + 03f145.63E + 031.17E + 042.16E + 038.95E + 039.23E + 03f155.56E + 031.17E + 041.66E + 039.04E + 039.26E + 03f164.88E + 031.16E + 041.10E + 038.10E + 038.34E + 03f175.19E + 031.15E + 041.33E + 038.44E + 038.65E + 03f184.31E + 038.92E + 031.60E + 038.06E + 038.30E + 03f194.84E + 031.16E + 042.00E + 038.77E + 038.92E + 03f205.71E + 038.46E + 031.58E + 039.09E + 039.35E + 03Table 10
Experimental results of the evaluation indicator (mean).
MeanTest1Test2Test3Test4Test5f111.12E + 031.16E + 031.15E + 031.11E + 031.11E + 03f122.99E + 052.70E + 063.01E + 053.22E + 052.90E + 05f136.66E + 038.66E + 036.77E + 035.33E + 035.19E + 03f141.47E + 031.48E + 031.48E + 031.43E + 031.43E + 03f151.96E + 031.81E + 032.24E + 031.66E + 031.66E + 03f161.63E + 031.64E + 031.68E + 031.61E + 031.61E + 03f171.72E + 031.75E + 031.74E + 031.72E + 031.72E + 03f186.62E + 031.18E + 048.50E + 038.50E + 038.14E + 03f192.05E + 032.05E + 032.58E + 031.93E + 031.93E + 03f202.02E + 032.04E + 032.04E + 032.02E + 032.02E + 03It can be seen from the Table9 that the diversity of Test2 for most functions other than f20 was the best among all the experiments, which shown that the simultaneous use of sorting grouping selection operator, mutation operator and catastrophe mechanism can improve population diversity. Knowing from the comparison results of Test1, Test3 and Test4 with Test5 respectively, in improving population diversity, mutation operator was better than sorting grouping selection operator, and sorting grouping selection operator was superior to catastrophe mechanism. From Table 10, we can see that Test5 has obtained the best experimental results, which proved that SGMOGA using all operators at the same time has the best performance and transposons crossover operator has strong search capability.


### 4.4.2. The Average Optimal Value, Standard Deviation and the Number of Times When Converging to the Global Optimal Solution

To obtain a fair performance comparison, 29 test functions of CEC2017 were used as examples and each test function was carried out 50 times with MATLAB R2016a. The experimental evaluation indicators were the average optimal value, minimum, maximum, standard deviation, and the number of times when converging to the global optimal solution under certain iterations, which can be represented respectively by avg, min, max, std and nos. The experimental results off1∼f10 for 2D problems are shown in Table 11. The experimental results of f11∼f20 for 10D problems and f21∼f30 for 30D problems at 10000 iterations are shown in Table 12.Table 11
The experimental results off1∼f10 for 2D problems.
f/genAlgorithmNosAvgMaxStdf1/10000SGA21.4273E + 022.5838E + 026.4120E + 01S-MOGA31.4515E + 022.5797E + 026.3623E + 01T-MOGA11.4816E + 022.5836E + 026.6638E + 01SGMOGA351.0000E + 021.0009E + 021.6911E − 02f3/100SGA143.1565E + 023.6039E + 022.1979E + 01S-MOGA203.1239E + 023.5893Ev022.0929E + 01T-MOGA213.1239E + 023.5891Ev022.0924E + 01SGMOGA503.0000E + 023.0000E + 022.4198E − 06f4/50SGA54.0002E + 024.0014E + 023.3144E − 02S-MOGA114.0001E + 024.0006E + 022.0530E − 02T-MOGA134.0001E + 024.0006E + 022.0391E − 02SGMOGA454.0000E + 024.0001E + 028.0654E − 04f5/100SGA105.0024E + 025.0100E + 024.1927E − 01S-MOGA195.0016E + 025.0116E + 023.7238E − 01T-MOGA205.0014E + 025.0145E + 023.7022E − 01SGMOGA415.0000E + 025.0000E + 021.0817E − 03f6/500SGA156.0023E + 026.0053E + 022.5440E − 01S-MOGA186.0021E + 026.0002E + 022.4970E − 01T-MOGA166.0019E + 026.0053E + 022.4387E − 01SGMOGA486.0000E + 026.0000E + 024.4401E − 05f7/500SGA37.0183E + 027.0304E + 028.6301E − 01S-MOGA27.0160E + 027.0263E + 028.5167E − 01T-MOGA37.0155E + 027.0298E + 029.2149E − 01SGMOGA507.0000E + 027.0000E + 026.2974E − 09f8/500SGA148.0065E + 028.0398E + 028.8120E − 01S-MOGA20 8.0042E + 028.0199E + 025.1377E − 01T-MOGA178.0048E + 028.0398E + 027.2909E − 01SGMOGA478.0000E + 028.0000E + 021.3008E − 04f9/100SGA129.0020E + 029.0095E + 022.7031E − 01S-MOGA109.0018E + 029.0065E + 022.5197E − 01T-MOGA149.0012E + 029.0065E + 022.1634E − 01SGMOGA359.0000E + 029.0010E + 021.3517E − 02f10/1000SGA21.0067E + 031.0171E + 038.0015E + 00S-MOGA31.0054E + 031.0171E + 037.5548E + 00T-MOGA61.0049E + 031.0171E + 037.4838E + 00SGMOGA281.0000E + 031.0002E + 033.4186E − 02Table 12
The experimental results off11∼f20 for 10D problems and f21∼f30 for 30D problems.
fAlgorithmAvgMinMaxStdf11SGA1.1294E + 031.1074E + 031291.0741675.3684E + 01S-MOGA1.1379E + 031.1043E + 031296.2561086.4838E + 01T-MOGA1.1092E + 031.1018E + 031119.5695935.0203E + 00SGMOGA1.1078E + 031.1021E + 031116.7007463.9382E + 00f12SGA1.4350E + 057.1900E + 039.4500E + 052.4850E + 05S-MOGA1.2578E + 057.9300E + 031.2000E + 062.6381E + 05T-MOGA2.2080E + 052.6400E + 031.8500E + 065.2809E + 05SGMOGA1.9592E + 042.1400E + 036.5000E + 041.8463E + 04f13SGA1.0082E + 041.3300E + 034.5200E + 041.0974E + 04S-MOGA7.7692E + 031.3200E + 032.0400E + 045.3334E + 03T-MOGA4.7878E + 031.3200E + 031.3700E + 043.9643E + 03SGMOGA1.5631E + 031.3200E + 032.1600E + 032.1820E + 02f14SGA2.0156E + 031.4084E + 035.0965E + 031.2556E + 03S-MOGA1.5349E + 031.4028E + 032.4366E + 032.2322E + 02T-MOGA1.4824E + 031.4039E + 031.6566E + 038.0107E + 01SGMOGA1.4130E + 031.4030E + 031.4387E + 038.3338E + 00f15SGA1.8449E + 031.5066E + 034.2233E + 036.0847E + 02S-MOGA1.7632E + 031.5034E + 034.2166E + 035.9500E + 02T-MOGA1.9437E + 031.5017E + 034.9532E + 039.0751E + 02SGMOGA1.5868E + 031.5019E + 031.9327E + 031.4655E + 02f16SGA1.6189E + 031.6005E + 031.7465E + 033.9286E + 01S-MOGA1.6235E + 031.6010E + 031.7477E + 034.7449E + 01T-MOGA1.6190E + 031.6004E + 031.7327E + 033.6837E + 01SGMOGA1.6016E + 031.6001E + 031.6129E + 032.6508E + 00f17SGA1.7097E + 031.7025E + 031.7342E + 037.6515E + 00S-MOGA1.7096E + 031.7017E + 031.7234E + 035.9522E + 00T-MOGA1.7093E + 031.7021E + 031.7369E + 038.8750E + 00SGMOGA1.7079E + 031.7012E + 031.7189E + 036.8870E + 00f18SGA4.4026E + 031.8812E + 031.2589E + 042.5799E + 03S-MOGA4.0847E + 031.8507E + 031.2810E + 042.5931E + 03T-MOGA4.8871E + 031.8531E + 031.2513E + 042.9346E + 03SGMOGA2.9571E + 031.8126E + 038.9009E + 031.9532E + 03f19SGA1.9159E + 031.9025E + 031.9446E + 031.5818E + 01S-MOGA1.9208E + 031.9017E + 031.9497E + 031.8795E + 01T-MOGA1.9203E + 031.9017E + 031.9635E + 032.0544E + 01SGMOGA1.9076E + 031.9016E + 031.9433E + 038.7784E + 00f20SGA2.0088E + 032.0013E + 032.0228E + 036.9580E + 00S-MOGA2.0095E + 032.0012E + 032.0244E + 036.6974E + 00T-MOGA2.0077E + 032.0012E + 032.0198E + 035.6305E + 00SGMOGA2.0037E + 032.0006E + 032.0204E + 034.1985E + 00f21SGA2.4831E + 032.4513E + 032.5066E + 031.2831E + 01S-MOGA2.4755E + 032.4384E + 032.4919E + 031.3226E + 01T-MOGA2.4787E + 032.4530E + 032.5062E + 031.4300E + 01SGMOGA2.3935E + 032.3598E + 032.4268E + 031.7609E + 01f22SGA4.2946E + 032.6184E + 038.3313E + 032.4313E + 03S-MOGA3.4246E + 032.5458E + 037.7281E + 031.7958E + 03T-MOGA2.8843E + 032.5425E + 037.4651E + 031.0524E + 03SGMOGA2.5643E + 032.4235E + 032.7233E + 039.5558E + 01f23SGA2.8492E + 032.8286E + 032.8644E + 031.1295E + 01S-MOGA2.8319E + 032.8069E + 032.8582E + 031.3322E + 01T-MOGA2.8304E + 032.8026E + 032.8600E + 031.7635E + 01SGMOGA2.7506E + 032.7141E + 032.7933E + 032.0916E + 01f24SGA3.0336E + 033.0096E + 033.0610E + 031.2011E + 01S-MOGA3.0242E + 033.0085E + 033.0558E + 031.1581E + 01T-MOGA3.0246E + 033.0399E + 033.0446E + 031.3840E + 01SGMOGA2.9326E + 032.8895E + 032.9995E + 033.0003E + 01f25SGA3.0475E + 032.9993E + 033.1783E + 033.9304E + 01S-MOGA3.0352E + 032.9797E + 033.1351E + 034.1771E + 01T-MOGA3.0367E + 032.9838E + 033.1921E + 035.3587E + 01SGMOGA2.9830E + 032.9366E + 033.0328E + 033.1622E + 01f26SGA5.7855E + 035.5328E + 036.0841E + 031.3774E + 02S-MOGA5.6551E + 035.4031E + 035.8844E + 031.5856E + 02T-MOGA5.4773E + 033.6355E + 035.9961E + 034.6909E + 02SGMOGA4.7061E + 033.5117E + 035.3600E + 034.4765E + 02f27SGA3.2528E + 033.2433E + 033.2592E + 034.5962E + 00S-MOGA3.2448E + 033.2101E + 033.2633E + 031.3109E + 01T-MOGA3.2427E + 033.2210E + 033.2607E + 031.0229E + 01SGMOGA3.2341E + 033.2089E + 033.2623E + 031.5596E + 01f28SGA3.4106E + 033.3169E + 033.4993E + 034.7750E + 01S-MOGA3.3893E + 033.3368E + 033.5259E + 034.6290E + 01T-MOGA3.3802E + 033.3167E + 033.4629E + 033.0698E + 01SGMOGA3.3963E + 033.2987E + 033.5851E + 038.6236E + 01f29SGA3.7879E + 033.5729E + 034.0574E + 031.4114E + 02S-MOGA3.6960E + 033.5055E + 033.9562E + 031.4737E + 02T-MOGA3.7258E + 033.5218E + 033.9474E + 031.3105E + 02SGMOGA3.6926E + 033.3526E + 033.9317E + 031.6212E + 02f30SGA2.7420E + 061.1300E + 052.2400E + 075.6133E + 06S-MOGA2.1569E + 068.7500E + 041.1700E + 073.1870E + 06T-MOGA8.8745E + 059.4700E + 046.0200E + 061.4033E + 06SGMOGA1.7635E + 052.4500E + 046.0500E + 051.3356E + 05As shown in Table11, the number of times that SGMOGA has achieved the global optimal solution was obviously more than that achieved by SGA, S-MOGA and T-MOGA under the same iterations. After the same amount of iterations, all four algorithms have reached the theoretically optimal solution at least once, so that no list the indicator (min) in Table 11. SGMOGA not only outperformed other algorithms on the other three indicators (avg), (max) and (std), but also the average optimal value accuracy of SGMOGA was less than 10–2 that other algorithms have not reached.It can be seen from Table12 that the average optimal value of 19 functions except f28 in SGMOGA were better than other algorithms. In indicator (min), SGMOGA was superior to other algorithms in 17 functions apart from f11, f14 and f15. The indicator (max) of SGMOGA on 17 functions except f20, f27 and f28 have surpass other algorithms. About standard deviation, SGMOGA to excel others on 12 functions.To better analyze the performance of the algorithm, we used boxplot to represent the statistical results of each algorithm for test problem by Figure6. Boxplot is an important tool for statistical analysis in the field of economics. It can well reflect the statistical distribution of data. As depicted in Figure 6, the results of SGMOGA on test functions were more concentrated and close to the optimal than the comparison algorithm, which shown that SGMOGA has better search performance and robustness.Figure 6
Boxplots of the results obtained by four algorithms. (a)f11 (b) f12 (c) f13 (d) f14 (e) f15 (f) f16 (g) f17 (h) f18 (i) f19 (j) f20 (k) f21 (l) f22.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)


### 4.4.3. Significance Detection

T-test (at significance level of 0.05) was carried out by using the average optimal value and standard deviation to verify whether there is significant difference between improved algorithm and comparison algorithm. The T-test results are shown in Table 13. Symbol “s” in the Table 13 represent the result of T-test. Specifically, “+” indicates that the performance of SGMOGA is significantly better than the comparison algorithm; “−” expresses SGMOA is significantly worse than comparison algorithm; “=” means that the two algorithms have the same performance. B/W/E respectively indicates the number of functions that SGMOGA has significantly better, worse and equivalent than the comparison algorithm.Table 13
The results ofT-test.
fSGAS-MOGAT-MOGAtpstpstpsf1−4.71220.0000+−5.01800.0000+−5.11030.0000+f3−5.03490.0000+−4.18610.0000+−4.18710.0000+f4−4.26560.0000+−3.44160.0004+−3.46500.0004+f5−4.04760.0001+−3.03820.0015+−2.67390.0044+f6−6.39290.0000+−5.94680.0000+−5.50910.0000+f7−14.99410.0000+−13.28410.0000+−11.89390.0000+f8−5.21580.0000+−5.78050.0000+−4.65530.0000+f9−5.22530.0000+−5.04410.0000+−3.91460.0001+f10−5.92090.0000+−5.05420.0000+−4.62970.0000+f11−2.83750.0028+−3.27660.0007+−1.55150.0620=f12−3.51610.0003+−2.83930.0027+−2.69250.0042+f13−5.48800.0000+−8.22120.0000+−5.74320.0000+f14−3.39350.0005+−3.85880.0001+−6.09310.0000+f15−2.91600.0022+−2.03550.0222+−2.74530.0036+f16−3.10680.0012+−3.25860.0008+−3.33140.0006+f17−1.23640.1096=−1.32060.0949=−0.88120.1902=f18−3.15870.0011+−2.45600.0079+−3.87130.0001+f19−3.24420.0008+−4.49950.0000+−4.01960.0001+f20−4.43760.0000+−5.18840.0000+−4.02710.0001+f21−29.07890.0000+−26.32850.0000+−26.55850.0000+f22−5.02840.0000+−3.38270.0005+−2.14130.0174+f23−29.33030.0000+−23.18220.0000+−20.62530.0000+f24−22.09850.0000+−20.13990.0000+−19.68870.0000+f25−9.04110.0000+−7.04540.0000+−6.10270.0000+f26−16.29620.0000+−14.13020.0000+−8.41010.0000+f27−8.13260.0000+−3.71370.0002+−3.26040.0008+f28−1.02580.1538=0.50570.3071=1.24370.1083=f29−3.13500.0011+−0.10970.4564=−1.12610.1314=f30−3.23100.0008+−4.39040.0000+−3.56700.0003+B/W/E27/0/226/0/325/0/4According to theT-test results in Table 13, at the significance level 0.05, SGMOGA was significantly better than SGA in 27 functions, remarkably superior to S-MOGA in 26 functions and markedly better than T-MOGA in 25 functions. And it can be seen that the performance of SGMOGA was almost the same as that of the comparison algorithm in 2, 3 and 4 functions, respectively. There was not functions significantly weaker than the comparison algorithms. Compared with SGA, S-MOGA and T-MOGA, SGMOGA shown better comprehensive performance and it was more stable.


### 4.4.4. The Average Optimal Value under Different Iterations

To further illustrate that SGMOGA proposed in this paper improved the search performance of genetic algorithms, the seven test functions were operated respectively, and the average optimal value under different iterations was obtained to verify the search efficiency of the improved algorithm. In order to compare the convergence of four algorithms more intuitively, Figures7–13 described the convergence curves of the four algorithms on seven test functions. As shown in Figures 7–13, it was obvious that the convergence speed of SGMOGA was significantly faster than the comparison algorithm and that SGMOGA proposed in this paper improved the search performance compared with SGA, S-MOGA and T-MOGA.Figure 7
The average optimal value of functionf1.Figure 8
The average optimal value of functionf3.Figure 9
The average optimal value of functionf4.Figure 10
The average optimal value of functionf6.Figure 11
The average optimal value of functionf7.Figure 12
The average optimal value of functionf8.Figure 13
The average optimal value of functionf10.


### 4.4.5. The Average Running Time of Iterations

To verify the convergence rate of the algorithm, the average running time of iterations when converging to the optimal solution were calculated as follows: when obtaining the calculation an accuracy of10−4, the running time of iterations at the ith calculation is t(i), i = 1, 2, …, N, where N is the number of runs. Then, the average running time of iterations can be calculated by :(13)tav=1N∑1Nti,where tav is the average running time of iterations.The average running time of iterations converging to the optimal solution were shown in Table14.Table 14
Average running time of iterations.
fSGAS-MOGAT-MOGASGMOGAf183.2110128.9689106.135817.8289f39.48228.27097.21730.1726f437.573536.226128.85670.1297f536.696037.399241.92720.1971f625.150651.265432.205210.6908f752.702947.575445.08730.3680f83.14811.51982.41270.1230f951.182336.930049.09240.2491f1092.3668108.6970128.41598.4093According to the experimental results in Table14, SGMOA reduced the average running time by 78.57%, 98.18%, 99.65%, 99.46%, 57.49%, 99.30%, 96.09%, 99.51% and 90.90% in comparison to SGA. Compared with S-MOGA, the average running time decrease rate were 86.18%, 97.91%, 99.64%, 99.47%, 79.15%, 99.23%, 91.91%, 99.33% and 92.26%. Moreover, this indicator was down by 83.20%, 97.61%, 99.55%, 99.53%, 66.80%, 99.18%, 94.90%, 99.49% and 93.45% relative to T-MOGA. In terms of the average running time when converging to the optimal solution, compared with SGA, S-MOGA and T-MOGA, the decline range of SGMOGA for f1∼f10 was 57.49%∼99.65%, 79.15%∼99.64% and 66.80%∼99.55% respectively. And this made the point that the average running time of SGMOGA was significantly superior to SGA, S-MOGA and T-MOGA. The results shown that SGMOGA has faster convergence to the global optimal solution.


### 4.4.6. Performance of the Proposed Method with Different Parameters

To verify the performance of the proposed method under different population sizes, and crossover and mutation values of the GA, experiments were conducted with one parameter changed and the others parameters unchanged. The values of unchanged parameters were the same as that of parameters for initialization above. The parameter values of the experiment were as follows: (1) population sizen: 50, 100, 150, 200, 300; (2) crossover value pc: 0.6, 0.7, 0.8, 0.9, 1; and (3) mutation value pm: 0.01, 0.05, 0.1, 0.15, 0.2.Takingf3 as an example, the average running time of iterations converging to the optimal solution under different population sizes, crossover values and mutation values were shown in Tables 15–17, respectively.Table 15
Average running time under different population sizes.
nSGAS-MOGAT-MOGASGMOGA509.72886.61955.66140.18151003.79883.30684.91600.17261505.62503.23452.52990.30682002.57773.01583.43420.36863003.43143.77993.80530.4774Table 16
Average running time under different crossover values.
pcSGAS-MOGAT-MOGASGMOGA0.66.80856.17275.36450.20780.75.52485.77244.37290.20320.85.99305.23464.29360.19360.93.79883.30684.91600.17261.07.14948.44085.24620.2479Table 17
Average running time under different mutation values.
pmSGAS-MOGAT-MOGASGMOGA0.0141.997147.373356.14100.28350.0531.501724.483330.11210.20380.103.79883.30684.91600.17260.151.00940.69630.90940.24260.201.03400.85111.04270.2547Thus, it was verified that the performance of SGMOGA with different population sizes and crossover and mutation values was superior to the performances of SGA, S-MOGA and T-MOGA. From the effects of GA parameters on the performance of the proposed method, SGMOGA was shown to have the best performance with population size equal to 100, crossover value equal to 0.9 and mutation value equal to 0.1.As the above experimental series and analysis showed, the proposed SGMOGA algorithm is an effective algorithm for function optimization. The reasons are concluded as follows. First, sorting grouping selection and combination paring operations are used before crossover, which easily implemented and enhanced the diversity of the population. Second, different from traditional MOGA, the mulit-offspring generated by transposons crossover operator in different pairs of parents instead of repeat crossover in same pair of parents, to avoid inbreeding and premature convergence. In addition, the use of transposons crossover operator would be conducive to transfer excellent genes to the next generation and accelerate convergence. Third, at the same time of local search using mutation operator, to avoid falling into local convergence, catastrophe mechanism is introduced into this algorithm to make the population restart diversified search, to find the global optimal solution. Finally, the strategy of elitism and population competition are used at the same time, to protect excellent individuals and enhance the guidance of the algorithm. Moreover, it can avoid the expansion of population size to affect the operation efficiency of SGMOGA algorithm. Therefore, the SGMOGA algorithm, which is supposed to take advantage of previous knowledge of MOGA, the theory of transposons and catastrophe, the strategy of elitism and population competition to create a new variant, is an effective GA algorithm for function optimization.


## 4.4.1. Verification of Operator Basic Functions

In order to study the role of each operator in SGMOGA, we disabled each operator in turn, and other operators were used normally. The algorithm that disabling sorting grouping selection operator but using roulette selection was marked as Test1, and disabling the transposons crossover operator, mutation operator, and catastrophe mechanism were recorded as Test2, Test3, and Test4, respectively. SGMOGA that using all operators at the same time was denoted by Test5. The experiment selectedf11∼f20 from the CEC2017 for testing. The evaluation indicators were the average population diversity (div) [48] and the average value (mean) of the optimal value. The results were shown in Tables 9 and 10.Table 9
Experimental results of the evaluation indicator (div).
DivTest1Test2Test3Test4Test5f115.26E + 031.14E + 049.07E + 028.82E + 039.04E + 03f124.18E + 031.13E + 048.46E + 027.35E + 037.74E + 03f135.19E + 031.16E + 041.66E + 038.38E + 038.65E + 03f145.63E + 031.17E + 042.16E + 038.95E + 039.23E + 03f155.56E + 031.17E + 041.66E + 039.04E + 039.26E + 03f164.88E + 031.16E + 041.10E + 038.10E + 038.34E + 03f175.19E + 031.15E + 041.33E + 038.44E + 038.65E + 03f184.31E + 038.92E + 031.60E + 038.06E + 038.30E + 03f194.84E + 031.16E + 042.00E + 038.77E + 038.92E + 03f205.71E + 038.46E + 031.58E + 039.09E + 039.35E + 03Table 10
Experimental results of the evaluation indicator (mean).
MeanTest1Test2Test3Test4Test5f111.12E + 031.16E + 031.15E + 031.11E + 031.11E + 03f122.99E + 052.70E + 063.01E + 053.22E + 052.90E + 05f136.66E + 038.66E + 036.77E + 035.33E + 035.19E + 03f141.47E + 031.48E + 031.48E + 031.43E + 031.43E + 03f151.96E + 031.81E + 032.24E + 031.66E + 031.66E + 03f161.63E + 031.64E + 031.68E + 031.61E + 031.61E + 03f171.72E + 031.75E + 031.74E + 031.72E + 031.72E + 03f186.62E + 031.18E + 048.50E + 038.50E + 038.14E + 03f192.05E + 032.05E + 032.58E + 031.93E + 031.93E + 03f202.02E + 032.04E + 032.04E + 032.02E + 032.02E + 03It can be seen from the Table9 that the diversity of Test2 for most functions other than f20 was the best among all the experiments, which shown that the simultaneous use of sorting grouping selection operator, mutation operator and catastrophe mechanism can improve population diversity. Knowing from the comparison results of Test1, Test3 and Test4 with Test5 respectively, in improving population diversity, mutation operator was better than sorting grouping selection operator, and sorting grouping selection operator was superior to catastrophe mechanism. From Table 10, we can see that Test5 has obtained the best experimental results, which proved that SGMOGA using all operators at the same time has the best performance and transposons crossover operator has strong search capability.


## 4.4.2. The Average Optimal Value, Standard Deviation and the Number of Times When Converging to the Global Optimal Solution

To obtain a fair performance comparison, 29 test functions of CEC2017 were used as examples and each test function was carried out 50 times with MATLAB R2016a. The experimental evaluation indicators were the average optimal value, minimum, maximum, standard deviation, and the number of times when converging to the global optimal solution under certain iterations, which can be represented respectively by avg, min, max, std and nos. The experimental results off1∼f10 for 2D problems are shown in Table 11. The experimental results of f11∼f20 for 10D problems and f21∼f30 for 30D problems at 10000 iterations are shown in Table 12.Table 11
The experimental results off1∼f10 for 2D problems.
f/genAlgorithmNosAvgMaxStdf1/10000SGA21.4273E + 022.5838E + 026.4120E + 01S-MOGA31.4515E + 022.5797E + 026.3623E + 01T-MOGA11.4816E + 022.5836E + 026.6638E + 01SGMOGA351.0000E + 021.0009E + 021.6911E − 02f3/100SGA143.1565E + 023.6039E + 022.1979E + 01S-MOGA203.1239E + 023.5893Ev022.0929E + 01T-MOGA213.1239E + 023.5891Ev022.0924E + 01SGMOGA503.0000E + 023.0000E + 022.4198E − 06f4/50SGA54.0002E + 024.0014E + 023.3144E − 02S-MOGA114.0001E + 024.0006E + 022.0530E − 02T-MOGA134.0001E + 024.0006E + 022.0391E − 02SGMOGA454.0000E + 024.0001E + 028.0654E − 04f5/100SGA105.0024E + 025.0100E + 024.1927E − 01S-MOGA195.0016E + 025.0116E + 023.7238E − 01T-MOGA205.0014E + 025.0145E + 023.7022E − 01SGMOGA415.0000E + 025.0000E + 021.0817E − 03f6/500SGA156.0023E + 026.0053E + 022.5440E − 01S-MOGA186.0021E + 026.0002E + 022.4970E − 01T-MOGA166.0019E + 026.0053E + 022.4387E − 01SGMOGA486.0000E + 026.0000E + 024.4401E − 05f7/500SGA37.0183E + 027.0304E + 028.6301E − 01S-MOGA27.0160E + 027.0263E + 028.5167E − 01T-MOGA37.0155E + 027.0298E + 029.2149E − 01SGMOGA507.0000E + 027.0000E + 026.2974E − 09f8/500SGA148.0065E + 028.0398E + 028.8120E − 01S-MOGA20 8.0042E + 028.0199E + 025.1377E − 01T-MOGA178.0048E + 028.0398E + 027.2909E − 01SGMOGA478.0000E + 028.0000E + 021.3008E − 04f9/100SGA129.0020E + 029.0095E + 022.7031E − 01S-MOGA109.0018E + 029.0065E + 022.5197E − 01T-MOGA149.0012E + 029.0065E + 022.1634E − 01SGMOGA359.0000E + 029.0010E + 021.3517E − 02f10/1000SGA21.0067E + 031.0171E + 038.0015E + 00S-MOGA31.0054E + 031.0171E + 037.5548E + 00T-MOGA61.0049E + 031.0171E + 037.4838E + 00SGMOGA281.0000E + 031.0002E + 033.4186E − 02Table 12
The experimental results off11∼f20 for 10D problems and f21∼f30 for 30D problems.
fAlgorithmAvgMinMaxStdf11SGA1.1294E + 031.1074E + 031291.0741675.3684E + 01S-MOGA1.1379E + 031.1043E + 031296.2561086.4838E + 01T-MOGA1.1092E + 031.1018E + 031119.5695935.0203E + 00SGMOGA1.1078E + 031.1021E + 031116.7007463.9382E + 00f12SGA1.4350E + 057.1900E + 039.4500E + 052.4850E + 05S-MOGA1.2578E + 057.9300E + 031.2000E + 062.6381E + 05T-MOGA2.2080E + 052.6400E + 031.8500E + 065.2809E + 05SGMOGA1.9592E + 042.1400E + 036.5000E + 041.8463E + 04f13SGA1.0082E + 041.3300E + 034.5200E + 041.0974E + 04S-MOGA7.7692E + 031.3200E + 032.0400E + 045.3334E + 03T-MOGA4.7878E + 031.3200E + 031.3700E + 043.9643E + 03SGMOGA1.5631E + 031.3200E + 032.1600E + 032.1820E + 02f14SGA2.0156E + 031.4084E + 035.0965E + 031.2556E + 03S-MOGA1.5349E + 031.4028E + 032.4366E + 032.2322E + 02T-MOGA1.4824E + 031.4039E + 031.6566E + 038.0107E + 01SGMOGA1.4130E + 031.4030E + 031.4387E + 038.3338E + 00f15SGA1.8449E + 031.5066E + 034.2233E + 036.0847E + 02S-MOGA1.7632E + 031.5034E + 034.2166E + 035.9500E + 02T-MOGA1.9437E + 031.5017E + 034.9532E + 039.0751E + 02SGMOGA1.5868E + 031.5019E + 031.9327E + 031.4655E + 02f16SGA1.6189E + 031.6005E + 031.7465E + 033.9286E + 01S-MOGA1.6235E + 031.6010E + 031.7477E + 034.7449E + 01T-MOGA1.6190E + 031.6004E + 031.7327E + 033.6837E + 01SGMOGA1.6016E + 031.6001E + 031.6129E + 032.6508E + 00f17SGA1.7097E + 031.7025E + 031.7342E + 037.6515E + 00S-MOGA1.7096E + 031.7017E + 031.7234E + 035.9522E + 00T-MOGA1.7093E + 031.7021E + 031.7369E + 038.8750E + 00SGMOGA1.7079E + 031.7012E + 031.7189E + 036.8870E + 00f18SGA4.4026E + 031.8812E + 031.2589E + 042.5799E + 03S-MOGA4.0847E + 031.8507E + 031.2810E + 042.5931E + 03T-MOGA4.8871E + 031.8531E + 031.2513E + 042.9346E + 03SGMOGA2.9571E + 031.8126E + 038.9009E + 031.9532E + 03f19SGA1.9159E + 031.9025E + 031.9446E + 031.5818E + 01S-MOGA1.9208E + 031.9017E + 031.9497E + 031.8795E + 01T-MOGA1.9203E + 031.9017E + 031.9635E + 032.0544E + 01SGMOGA1.9076E + 031.9016E + 031.9433E + 038.7784E + 00f20SGA2.0088E + 032.0013E + 032.0228E + 036.9580E + 00S-MOGA2.0095E + 032.0012E + 032.0244E + 036.6974E + 00T-MOGA2.0077E + 032.0012E + 032.0198E + 035.6305E + 00SGMOGA2.0037E + 032.0006E + 032.0204E + 034.1985E + 00f21SGA2.4831E + 032.4513E + 032.5066E + 031.2831E + 01S-MOGA2.4755E + 032.4384E + 032.4919E + 031.3226E + 01T-MOGA2.4787E + 032.4530E + 032.5062E + 031.4300E + 01SGMOGA2.3935E + 032.3598E + 032.4268E + 031.7609E + 01f22SGA4.2946E + 032.6184E + 038.3313E + 032.4313E + 03S-MOGA3.4246E + 032.5458E + 037.7281E + 031.7958E + 03T-MOGA2.8843E + 032.5425E + 037.4651E + 031.0524E + 03SGMOGA2.5643E + 032.4235E + 032.7233E + 039.5558E + 01f23SGA2.8492E + 032.8286E + 032.8644E + 031.1295E + 01S-MOGA2.8319E + 032.8069E + 032.8582E + 031.3322E + 01T-MOGA2.8304E + 032.8026E + 032.8600E + 031.7635E + 01SGMOGA2.7506E + 032.7141E + 032.7933E + 032.0916E + 01f24SGA3.0336E + 033.0096E + 033.0610E + 031.2011E + 01S-MOGA3.0242E + 033.0085E + 033.0558E + 031.1581E + 01T-MOGA3.0246E + 033.0399E + 033.0446E + 031.3840E + 01SGMOGA2.9326E + 032.8895E + 032.9995E + 033.0003E + 01f25SGA3.0475E + 032.9993E + 033.1783E + 033.9304E + 01S-MOGA3.0352E + 032.9797E + 033.1351E + 034.1771E + 01T-MOGA3.0367E + 032.9838E + 033.1921E + 035.3587E + 01SGMOGA2.9830E + 032.9366E + 033.0328E + 033.1622E + 01f26SGA5.7855E + 035.5328E + 036.0841E + 031.3774E + 02S-MOGA5.6551E + 035.4031E + 035.8844E + 031.5856E + 02T-MOGA5.4773E + 033.6355E + 035.9961E + 034.6909E + 02SGMOGA4.7061E + 033.5117E + 035.3600E + 034.4765E + 02f27SGA3.2528E + 033.2433E + 033.2592E + 034.5962E + 00S-MOGA3.2448E + 033.2101E + 033.2633E + 031.3109E + 01T-MOGA3.2427E + 033.2210E + 033.2607E + 031.0229E + 01SGMOGA3.2341E + 033.2089E + 033.2623E + 031.5596E + 01f28SGA3.4106E + 033.3169E + 033.4993E + 034.7750E + 01S-MOGA3.3893E + 033.3368E + 033.5259E + 034.6290E + 01T-MOGA3.3802E + 033.3167E + 033.4629E + 033.0698E + 01SGMOGA3.3963E + 033.2987E + 033.5851E + 038.6236E + 01f29SGA3.7879E + 033.5729E + 034.0574E + 031.4114E + 02S-MOGA3.6960E + 033.5055E + 033.9562E + 031.4737E + 02T-MOGA3.7258E + 033.5218E + 033.9474E + 031.3105E + 02SGMOGA3.6926E + 033.3526E + 033.9317E + 031.6212E + 02f30SGA2.7420E + 061.1300E + 052.2400E + 075.6133E + 06S-MOGA2.1569E + 068.7500E + 041.1700E + 073.1870E + 06T-MOGA8.8745E + 059.4700E + 046.0200E + 061.4033E + 06SGMOGA1.7635E + 052.4500E + 046.0500E + 051.3356E + 05As shown in Table11, the number of times that SGMOGA has achieved the global optimal solution was obviously more than that achieved by SGA, S-MOGA and T-MOGA under the same iterations. After the same amount of iterations, all four algorithms have reached the theoretically optimal solution at least once, so that no list the indicator (min) in Table 11. SGMOGA not only outperformed other algorithms on the other three indicators (avg), (max) and (std), but also the average optimal value accuracy of SGMOGA was less than 10–2 that other algorithms have not reached.It can be seen from Table12 that the average optimal value of 19 functions except f28 in SGMOGA were better than other algorithms. In indicator (min), SGMOGA was superior to other algorithms in 17 functions apart from f11, f14 and f15. The indicator (max) of SGMOGA on 17 functions except f20, f27 and f28 have surpass other algorithms. About standard deviation, SGMOGA to excel others on 12 functions.To better analyze the performance of the algorithm, we used boxplot to represent the statistical results of each algorithm for test problem by Figure6. Boxplot is an important tool for statistical analysis in the field of economics. It can well reflect the statistical distribution of data. As depicted in Figure 6, the results of SGMOGA on test functions were more concentrated and close to the optimal than the comparison algorithm, which shown that SGMOGA has better search performance and robustness.Figure 6
Boxplots of the results obtained by four algorithms. (a)f11 (b) f12 (c) f13 (d) f14 (e) f15 (f) f16 (g) f17 (h) f18 (i) f19 (j) f20 (k) f21 (l) f22.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)


## 4.4.3. Significance Detection

T-test (at significance level of 0.05) was carried out by using the average optimal value and standard deviation to verify whether there is significant difference between improved algorithm and comparison algorithm. The T-test results are shown in Table 13. Symbol “s” in the Table 13 represent the result of T-test. Specifically, “+” indicates that the performance of SGMOGA is significantly better than the comparison algorithm; “−” expresses SGMOA is significantly worse than comparison algorithm; “=” means that the two algorithms have the same performance. B/W/E respectively indicates the number of functions that SGMOGA has significantly better, worse and equivalent than the comparison algorithm.Table 13
The results ofT-test.
fSGAS-MOGAT-MOGAtpstpstpsf1−4.71220.0000+−5.01800.0000+−5.11030.0000+f3−5.03490.0000+−4.18610.0000+−4.18710.0000+f4−4.26560.0000+−3.44160.0004+−3.46500.0004+f5−4.04760.0001+−3.03820.0015+−2.67390.0044+f6−6.39290.0000+−5.94680.0000+−5.50910.0000+f7−14.99410.0000+−13.28410.0000+−11.89390.0000+f8−5.21580.0000+−5.78050.0000+−4.65530.0000+f9−5.22530.0000+−5.04410.0000+−3.91460.0001+f10−5.92090.0000+−5.05420.0000+−4.62970.0000+f11−2.83750.0028+−3.27660.0007+−1.55150.0620=f12−3.51610.0003+−2.83930.0027+−2.69250.0042+f13−5.48800.0000+−8.22120.0000+−5.74320.0000+f14−3.39350.0005+−3.85880.0001+−6.09310.0000+f15−2.91600.0022+−2.03550.0222+−2.74530.0036+f16−3.10680.0012+−3.25860.0008+−3.33140.0006+f17−1.23640.1096=−1.32060.0949=−0.88120.1902=f18−3.15870.0011+−2.45600.0079+−3.87130.0001+f19−3.24420.0008+−4.49950.0000+−4.01960.0001+f20−4.43760.0000+−5.18840.0000+−4.02710.0001+f21−29.07890.0000+−26.32850.0000+−26.55850.0000+f22−5.02840.0000+−3.38270.0005+−2.14130.0174+f23−29.33030.0000+−23.18220.0000+−20.62530.0000+f24−22.09850.0000+−20.13990.0000+−19.68870.0000+f25−9.04110.0000+−7.04540.0000+−6.10270.0000+f26−16.29620.0000+−14.13020.0000+−8.41010.0000+f27−8.13260.0000+−3.71370.0002+−3.26040.0008+f28−1.02580.1538=0.50570.3071=1.24370.1083=f29−3.13500.0011+−0.10970.4564=−1.12610.1314=f30−3.23100.0008+−4.39040.0000+−3.56700.0003+B/W/E27/0/226/0/325/0/4According to theT-test results in Table 13, at the significance level 0.05, SGMOGA was significantly better than SGA in 27 functions, remarkably superior to S-MOGA in 26 functions and markedly better than T-MOGA in 25 functions. And it can be seen that the performance of SGMOGA was almost the same as that of the comparison algorithm in 2, 3 and 4 functions, respectively. There was not functions significantly weaker than the comparison algorithms. Compared with SGA, S-MOGA and T-MOGA, SGMOGA shown better comprehensive performance and it was more stable.


## 4.4.4. The Average Optimal Value under Different Iterations

To further illustrate that SGMOGA proposed in this paper improved the search performance of genetic algorithms, the seven test functions were operated respectively, and the average optimal value under different iterations was obtained to verify the search efficiency of the improved algorithm. In order to compare the convergence of four algorithms more intuitively, Figures7–13 described the convergence curves of the four algorithms on seven test functions. As shown in Figures 7–13, it was obvious that the convergence speed of SGMOGA was significantly faster than the comparison algorithm and that SGMOGA proposed in this paper improved the search performance compared with SGA, S-MOGA and T-MOGA.Figure 7
The average optimal value of functionf1.Figure 8
The average optimal value of functionf3.Figure 9
The average optimal value of functionf4.Figure 10
The average optimal value of functionf6.Figure 11
The average optimal value of functionf7.Figure 12
The average optimal value of functionf8.Figure 13
The average optimal value of functionf10.


## 4.4.5. The Average Running Time of Iterations

To verify the convergence rate of the algorithm, the average running time of iterations when converging to the optimal solution were calculated as follows: when obtaining the calculation an accuracy of10−4, the running time of iterations at the ith calculation is t(i), i = 1, 2, …, N, where N is the number of runs. Then, the average running time of iterations can be calculated by :(13)tav=1N∑1Nti,where tav is the average running time of iterations.The average running time of iterations converging to the optimal solution were shown in Table14.Table 14
Average running time of iterations.
fSGAS-MOGAT-MOGASGMOGAf183.2110128.9689106.135817.8289f39.48228.27097.21730.1726f437.573536.226128.85670.1297f536.696037.399241.92720.1971f625.150651.265432.205210.6908f752.702947.575445.08730.3680f83.14811.51982.41270.1230f951.182336.930049.09240.2491f1092.3668108.6970128.41598.4093According to the experimental results in Table14, SGMOA reduced the average running time by 78.57%, 98.18%, 99.65%, 99.46%, 57.49%, 99.30%, 96.09%, 99.51% and 90.90% in comparison to SGA. Compared with S-MOGA, the average running time decrease rate were 86.18%, 97.91%, 99.64%, 99.47%, 79.15%, 99.23%, 91.91%, 99.33% and 92.26%. Moreover, this indicator was down by 83.20%, 97.61%, 99.55%, 99.53%, 66.80%, 99.18%, 94.90%, 99.49% and 93.45% relative to T-MOGA. In terms of the average running time when converging to the optimal solution, compared with SGA, S-MOGA and T-MOGA, the decline range of SGMOGA for f1∼f10 was 57.49%∼99.65%, 79.15%∼99.64% and 66.80%∼99.55% respectively. And this made the point that the average running time of SGMOGA was significantly superior to SGA, S-MOGA and T-MOGA. The results shown that SGMOGA has faster convergence to the global optimal solution.


## 4.4.6. Performance of the Proposed Method with Different Parameters

To verify the performance of the proposed method under different population sizes, and crossover and mutation values of the GA, experiments were conducted with one parameter changed and the others parameters unchanged. The values of unchanged parameters were the same as that of parameters for initialization above. The parameter values of the experiment were as follows: (1) population sizen: 50, 100, 150, 200, 300; (2) crossover value pc: 0.6, 0.7, 0.8, 0.9, 1; and (3) mutation value pm: 0.01, 0.05, 0.1, 0.15, 0.2.Takingf3 as an example, the average running time of iterations converging to the optimal solution under different population sizes, crossover values and mutation values were shown in Tables 15–17, respectively.Table 15
Average running time under different population sizes.
nSGAS-MOGAT-MOGASGMOGA509.72886.61955.66140.18151003.79883.30684.91600.17261505.62503.23452.52990.30682002.57773.01583.43420.36863003.43143.77993.80530.4774Table 16
Average running time under different crossover values.
pcSGAS-MOGAT-MOGASGMOGA0.66.80856.17275.36450.20780.75.52485.77244.37290.20320.85.99305.23464.29360.19360.93.79883.30684.91600.17261.07.14948.44085.24620.2479Table 17
Average running time under different mutation values.
pmSGAS-MOGAT-MOGASGMOGA0.0141.997147.373356.14100.28350.0531.501724.483330.11210.20380.103.79883.30684.91600.17260.151.00940.69630.90940.24260.201.03400.85111.04270.2547Thus, it was verified that the performance of SGMOGA with different population sizes and crossover and mutation values was superior to the performances of SGA, S-MOGA and T-MOGA. From the effects of GA parameters on the performance of the proposed method, SGMOGA was shown to have the best performance with population size equal to 100, crossover value equal to 0.9 and mutation value equal to 0.1.As the above experimental series and analysis showed, the proposed SGMOGA algorithm is an effective algorithm for function optimization. The reasons are concluded as follows. First, sorting grouping selection and combination paring operations are used before crossover, which easily implemented and enhanced the diversity of the population. Second, different from traditional MOGA, the mulit-offspring generated by transposons crossover operator in different pairs of parents instead of repeat crossover in same pair of parents, to avoid inbreeding and premature convergence. In addition, the use of transposons crossover operator would be conducive to transfer excellent genes to the next generation and accelerate convergence. Third, at the same time of local search using mutation operator, to avoid falling into local convergence, catastrophe mechanism is introduced into this algorithm to make the population restart diversified search, to find the global optimal solution. Finally, the strategy of elitism and population competition are used at the same time, to protect excellent individuals and enhance the guidance of the algorithm. Moreover, it can avoid the expansion of population size to affect the operation efficiency of SGMOGA algorithm. Therefore, the SGMOGA algorithm, which is supposed to take advantage of previous knowledge of MOGA, the theory of transposons and catastrophe, the strategy of elitism and population competition to create a new variant, is an effective GA algorithm for function optimization.


## 5. Application of Improved Algorithm in Location Selection of Agricultural Products Logistics Facilities

Logistics facility location is an important issue in enterprise logistics strategic planning, which affects the actual logistics operation efficiency and cost. In this paper, the improved algorithm was used to optimize the location of raw material purchasing stations of a sugar company. The sugar company purchased sugar beet from 97 raw material planting sites and adopted two-stage transportation mode. Firstly, the farmers transport the sugar beet to the corresponding purchasing station, and then the sugar factory arranges vehicles to transport the sugar beet from purchasing stations to the central station in sugar factory. The transportation rates of the two stages are 0.5 and 0.25 yuan per ton kilometer respectively. In order to optimize the location of purchasing stations, all planting sites have formed eight cluster regions by K-means clustered with SPSS 13.0. Due to the sugar company is located in the first cluster area, sugar beets can be transported directly to the central station by farmers in this area. The clustering results, coordinates and freight volume of various planting sites were shown in the Table18. The coordinate of the sugar factory was filled in the first row in Table 18. In this agricultural product logistics facility location problem, there was almost no warehousing cost since beets were stored in the field near the purchasing station after the purchase. It is necessary to optimize the location of beets purchasing stations in other seven cluster areas with the goal of minimizing the total transportation cost. The objective function can be expressed by :(14)C=∑i=1Mc1diwi+∑i=1M∑j=1Nc2dijwij,where C is the total transportation cost, c1 and c2 are the transportation rates of the two transportation stages respectively, di and wi are the euclidean distance and total transportation volume between the ith purchasing station and sugar company, dij and wij are the euclidean distance and transportation volume between the jth planting site in the ith cluster area and ith purchasing station.Table 18
The clustering result, coordinates and freight volume of various planting sites.
No(X, Y)VolumeNo.(X, Y)VolumeNo.(X, Y)Volume1(68.39,85.78)—(46.40,73.19)5567.55(60.83,48.75)1234.20(55.99,85.42)3373.50(30.12,60.23)812.55(64.66,51.27)1261.80(58.13,86.71)3629.55(29.17,64.09)1375.20(71.91,48.43)1290.45(63.72,82.08)2418.45(21.22,65.42)1483.05(70.08,53.13)1055.55(63.76,83.80)3542.25(23.63,70.92)1131.00(72.70,54.25)1517.55(62.99,87.14)3448.20(28.57,68.81)1424.257(51.99,73.55)2769.45(65.57,88.26)1911.00(34.32,73.19)3560.55(47.01,62.94)3868.95(58.86,90.36)6957.30(32.43,76.80)1218.75(50.84,65.29)6189.75(54.04,100.28)1890.00(26.38,78.52)1999.50(54.48,65.91)1976.55(55.16,96.33)1988.254(28.51,20.46)2611.20(57.46,64.29)1893.30(58.62,97.69)1267.80(24.48,24.90)1167.75(54.28,61.85)3115.50(63.03,98.33)2110.95(31.29,33.18)1360.95(56.62,60.08)2199.00(61.57,100.43)1897.05(20.84,38.97)736.80(58.08,58.42)1396.50(64.41,100.30)2671.05(32.92,39.17)2887.05(57.19,54.29)3399.45(71.17,99.87)1116.75(36.02,37.74)608.70(64.85,58.80)1816.95(69.78,98.32)787.20(41.43,34.53)1079.55(66.13,58.45)1933.05(74.94,97.99)1334.55(26.53,42.80)2705.70(61.78,65.78)5679.00(76.09,96.85)5401.20(30.13,45.78)2026.80(57.02,74.41)3165.302(62.51,109.10)680.55(36.76,43.41)1978.50(60.07,72.95)4688.25(65.12,115.95)733.80(30.38,49.63)1533.308(43.21,45.34)2149.05(65.78,110.81)2110.50(33.40,50.21)2034.30(45.67,45.86)1072.50(67.90,113.59)910.05(24.82,52.98)900.45(48.25,46.88)2584.80(67.45,118.15)1917.755(57.38,32.27)403.20(42.29,49.90)1295.25(69.13,111.71)2774.25(58.20,35.13)982.50(41.36,57.83)2193.30(70.44,114.08)4434.45(48.94,39.60)2605.50(45.85,57.02)2280.75(73.38,116.00)843.75(53.55,44.96)2262.45(49.10,59.73)2309.25(71.99,113.10)765.00(57.62,49.17)2578.20(44.69,61.78)2244.30(81.32,103.30)2161.50(45.02,41.16)1241.25(41.74,54.35)2644.953(38.02,64.36)825.006(68.10,38.35)2172.30(53.47,52.26)4027.50(41.97,72.41)3414.00(70.93,35.64)2139.30(48.52,53.08)1870.05(38.71,73.88)5205.75(71.01,40.20)1302.30(51.34,53.89)1451.25(41.28,69.20)1867.50(70.66,43.80)1143.00(38.34,49.56)1481.55(38.06,69.42)1266.75(62.42,43.03)3065.55In order to verify the effectiveness of the improved algorithm in the location problem of agricultural product logistics facilities, comparative experiments were carried out. The minimum (min), maximum (max), mean and standard deviation (std) of the objective function were taken as the experimental evaluation indexes. AndT-test (at significance level of 0.05) were carried out to verify the significant difference. The experimental results were shown in the Table 19. Taking the optimal value solved by improved algorithm as the final result, the coordinates of purchasing stations after optimization were shown in Table 20.Table 19
Experimental results of different algorithms.
IndexSGAS-MOGAT-MOGASGMOGAMin2505867.7052253437.7372253440.9002253398.151Max2506373.8512254880.6382254747.0812254312.006Mean2506160.5112253922.4232253921.9552253746.058Std147.1251435358.7977788297.4370977259.8507781T-testtptptp−46000.0000−2.18050.0166−2.43930.0089Table 20
Coordinates of purchasing stations after optimization.
StationCoordinateStationCoordinate1(68.39, 85.78)5(56.2651502, 48.57200485)2(69.1472139, 111.6366515)6(68.06084149, 50.68657741)3(42.10867927, 73.07995865)7(59.32009609, 70.88949015)4(34.14964894, 45.73223565)8(48.73985284, 57.66694247)It can be seen from the experimental results in Table19, the indexes of the improved algorithm SGMOGA in this paper were significantly better than the comparison algorithm, which shown that the improved algorithm can more effectively optimize the location problem of the agricultural product purchasing stations.


## 6. Conclusions and Discussion

In this paper, a multi-offspring genetic algorithm based on sorting grouping selection and combination pairing crossover was proposed to solve function optimization. First, verification experimental results shown that the simultaneous use of sorting grouping selection operator, mutation operator and catastrophe mechanism can improve population diversity, and proved that SGMOGA using all operators at the same time has the best performance and transposons crossover operator has strong search capability. Second, through the results of some indicators, boxplot andT-test, it can be found that SGMOGA has significant performance advantages and stable performance. Moreover, SGMOGA also shown good convergence performance and speed through the convergence curve and operation time results. After that, it was also verified that the performance of SGMOGA under different population sizes, crossover, and mutation values was superior to comparison algorithms. Therefore, SGMOGA is an effective method to solve function optimization. Finally, applying the improved algorithm to the location optimization problem of agricultural product logistics facilities, it shown that the improved algorithm is an effective method to solve the location optimization.There are many issues of SGMOGA that are worth further studying and perfecting. For example, the sorting method and grouping mode of individuals, the improvement of the crossover operator, and the relationship between the number of offspring and the calculation speed, among others, are potential future directions.



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*Source: 4203082-2022-03-28.xml*