I Introduction
As stochastic search methods based on population, evolutionary algorithms (EAs) have been applied successfully in many fields [1, 2]. Nevertheless, for nontrivial problems, EAs tend to premature convergence. Fortunately, using a parallel EA (PEA) often leads to a superior numerical performance because EAs are naturally prone to parallelism [1]. Among types of PEAs, distributed EAs (DEAs), which can be readily implemented in distributed memory MIMD computers [1], are most popular [1, 3].
In a DEA, a large population is divided into a number of subpopulations. The parallel operator, migration, exchanges individuals between subpopulations at intervals [4]. As a result, global convergence behavior is improved. The first idea of DEA can be traced back to [5]. So far, many papers focus on DEAs. In recent years, the application range of DEA is wider and wider for more and more suitable parallel computing environments being available. For instance, in [6, 7, 8, 9, 10], there are DEAs for problems of different fields. On such a background, it is meaningful to improve migration operator for enhancing DEAs. As one way for this purpose, diversity based migration has been studied in some papers (e.g. [11, 12, 13, 14]).
The main motivation of this paper is as follows. According to literature, diversity based migration do help DEAs to get better solutions. However, existing related studies only focus on selecting fit individuals for migration. In fact, diversity based migration may also be realized by selecting migration moments, which is determined only by the parameter, migration interval in traditional DEAs. Nevertheless, such a topic is rarely seen in literature.
A scheme of setting the success rate of migration based on subpopulation diversity at each interval is proposed in this paper for DEAs. The idea behind it is selecting migration moments based on not only interval but also subpopulation diversity. More precisely, migration still occurs at intervals, but the probability of immigrants entering the target subpopulation, , will be decided by the diversity of this subpopulation, , according to a formula building the relationship of them. In detail, at migration intervals, is computed first. Then, is calculated according to the formula. At the probability, , immigrants enter the subpopulation .
An experiment is conducted on eight instances of the Travelling Salesman Problem (TSP) from [15] to compare the outcomes of the DEA with the proposed scheme and those of a traditional DEA. The only difference between them is that the proposed scheme is used only in the former. For the formula in the scheme, two parameters, and , are required. In our experiment, the common parameters in both algorithms are set the same value. Moreover, the algorithm based on the scheme is tested under nine value combinations of and . Results show that the DEA based on our scheme can get better and more stable solutions than the traditional one. Further, under some value combinations of and , the advantage of the algorithm with our scheme is much more remarkable.
The rest of this paper is organized as follows. Related researches are described in Section 2. In Section 3, the proposed scheme is presented. Then, experimental results are shown and analyzed in Section 4. Finally, a conclusion and a prospect are dealt with in Section 5.
Ii Related Researches
Iia Distributed Evolutionary Algorithms
A DEA can be considered as the upgrade of a EA to enhance solving ability. In [4], it is expressed as the Algorithm 1 whose italic part is the migration operation.
Main parallel parameters in DEAs are introduced as below:

Migration strategy, it defines how to select emigrants and choice replaced individuals in each subpopulation.

Migration rate (“r”in Algorithm 1), it specifies the quantity of individuals which emigrate from each subpopulation in one migration round.

Subpopulation size, it is the quantity of individuals in each subpopulation.

Quantity of subpopulation (“n”in Algorithm 1), it is the number of subpopulations.
On one hand, semiisolation can maintain difference among subpopulations. On the other hand, for subpopulations, at intervals, migration is expected to afford individuals whose fitness are close to that of local ones but have some different building blocks [17]. After migration, immigrants participating in variation operations can help to resist premature convergence if they do have some new building blocks. In this way, stagnation is postponed.
IiB Diversity Based Migration in DEAs
Diversity refers to differences among individuals, which can be measured at the genotype or phenotype levels [18]. Thus, diversity of a colony shows the its convergence degree. Though diversity measures at the latter level often demands much less time consumption, they are not as accurate as ones at the former one because the difference on fitness is not equivalent to that on chromosome structure.
Some instances on diversity based migration from literature are listed as below. Their diversity measures are all at the genotype level. In [11], a weighted sum of fitness and diversity measure is used in a DEA for selecting emigrants and replaced individuals. Results showed a reduction in the time taken to find the optimal solution across a range of benchmark problems. In the DEA presented by [12], the subpopulation representative what is an individual having the lowest average of distance to other all individuals in this subpopulation is found by calculation. Then, it is selected to be the emigrant. Furthermore, if migration rate is set more than one, some other individuals are chosen as emigrants based on the distance from it. This strategy has been tested on a number of problems and has consistently outperformed the standard migration strategy. The scheme in [13] only allows different enough individuals migrate to a subpopulation in order to keeping diversity of this subpopulation. To realize this scheme, a method for evaluating the similarity of two subpopulations is presented. More recently, multikulti methods, are proposed in [14] for selecting individuals in source subpopulation different enough to ones in target subpopulation . In order to do this, information on the composition of source subpopulation is required when emigrants are selected. Different ways of providing this information in a concise manner are considered. The results of experiments prove the usefulness of the multikulti strategies. Besides, The success of this kind of strategies is explained via the measurement of entropy as a representation of population diversity. In conclusion, existing schemes for diversity based migration concentrate on selecting individuals for migration with the guide of diversity.
Iii Proposed Scheme Description
Iiia Preliminary
Migration may bring new building blocks for target subpopulation, while increases the similarity between source subpopulation and target one inevitably. Hence, there are many occasions not fit for migration during a run. For instance, in a subpopulation, migration may not be required when getting better individuals by variation operators is easy. In such a case, migration have little positive influence but still decreases the difference between subpopulations. Anyhow, migration moments predetermined by intervals cannot match the dynamic situation of evolution. Consequently, inappropriate migration cannot be avoided effectively.
IiiB Scheme to Solve Existing Problem
Our thinking is that more migration chances should be provided when subpopulation diversity is low. In general, the computation for subpopulation diversity is timeconsuming in most EAs. Consequently, it is not fit to decide migration moments only based on subpopulation diversity since, in such a plan, the computation for subpopulation diversity has to be executed frequently. Also, setting a value of subpopulation diversity as the threshold for migration is not adopted by us. Two of reasons are listed below. Firstly, the fitting value is difficult to find and may vary much from one occasion to another. Besides, migration may not occur at all in some periods, such as the initial stage.
In our scheme, the attempt of migration still occurs at intervals, but the probability of immigrants entering the target subpopulation, , will be computed according to the formula that builds the relationship between subpopulation diversity, , and the probability, . At first, the simplest formula to describe the relationship between and , Formula 1, is designed by us.
(1) 
In the initial stage of a run, the value of is close to the maximum, 1. According to Formula 1, is near to 0 at this time. Then, decreasing monotonically makes increasing monotonically. In the end, approaches 1 since has come to the minimum, 0. Then, Formula 2, where , and Formula 3, are considered.
(2) 
(3) 
Finally, Formula 4 is adopted since the three former ones can be regarded as special cases of it.
(4) 
Essentially, and are just used to adjust the relationship between and .
The flowchart of a DEA with our scheme is shown as Algorithm 2. The scheme is in the underlined and italic text of Algorithm 2.
Essentially, our scheme devotes to control the probability of completing migration according to subpopulation diversity at intervals rather than to improve subpopulation diversity.
IiiC Analysis of Our Scheme on Running Time
Provided that all supopulations in DEAs remains in exact synchronization, the consuming time for the two algorithms can be expressed as below, respectively. Let be the consuming time of all evolutionary operations in one generation, be that of once diversity computation, be that of a migration round, be interval and be total generations. In a DEA based on the proposed scheme, the total consuming time in a run, , can be computed as follow:
(5) 
In a DEA without the proposed scheme, be the consuming time of a migration round. Then, the total consuming time in a run, can be computed as follow:
(6) 
If above two types of DEA are based on the same EA and same in settings of common parameters, comparisons can be done as follow: Firstly,
(7) 
mainly because, in the DEA with the scheme, a migration round requires the maximum time, , only when is satisfied. Moreover, the total consuming time of the former algorithm have a peculiar part,
(8) 
Therefore, the difference of these two algorithms on total consuming time in a run can be measured as below:
(9) 
The total consuming time of the former is larger than that of the latter because
(10) 
That is, consuming time of once diversity computation is much more than the difference in that of a migration round for different algorithms.
In the scheme, subpopulation diversity is computed only at intervals. Although the diversity measure may vary according to chromosome coding, the extra consuming time in a DEA for the proposed scheme will always be acceptable if migration interval, , is large enough. In fact, to maintain the difference between subpopulations, is always large in the majority of DEAs. Therefore, this scheme can be widely used in DEAs.
Iv Experiment Studies
Iva Selected Problem and EA for It
The famous NPhard problem, TSP, is selected for our experiment. [15] has its benchmark instances. Some EA for the TSP have remarkable performance. For instance, the one presented in [19]
outperforms stateoftheart heuristic algorithms in finding very highquality solutions on instances with up to 200,000 cities.
It should be stressed that the purpose of our experiments is not to find better solutions of any problem than ever but to test our scheme by comparing the performance of a DEA with it and that of a traditional one. For this purpose, the more powerful a EA is, the larger instances, which demand much more on resource, should be used. In consideration of this, the EA in [20] proposed for years is used in our experiment. It is based on the inverover operator [21], the selection method that each individual competes with its offspring only and socalled mapping operator [20]. The flow of mapping is as follow. Firstly, two individuals are selected at random. Then, a segment of chromosome in the individual having worse fitness is selected at random. After that, a segment which has the same number of cities and the same first city is searched in the other individual. As soon as it is found, the former segment in the worse individual is replaced by it. Finally, in the worse individual, the other part of the chromosome is adjusted according to the latter steps of partially mapped crossover [22] which is a traditional operator used in EAs for the TSP. It can be seen from the flow that mapping belongs to crossover. In this EA, evolutionary velocity, , which is calculated as the Formula 11, decides whether the mapping operator should be executed.
(11) 
In this formula, is the fitness of the current best individual, is the that of the previous one and denotes the generations between the appearance of the previous best individual and that of the current one. Mapping is carried out only when is lower than a threshold value.
IvB Diversity Measure in Our DEA
Matrix M in Formula 12 is connection matrix of TSP tour [23].
(12) 
In the matrix, is the number of cities and . represents that there is a connection from city to city in tour, while denotes that such a connection does not exist. Let and be two individuals. Then, each of them has a connection matrix. Let be the number of rows which are same in the two matrixes. Then, the difference between and can be defined as Formula 13 [23].
(13) 
Then, in [23], subpopulation diversity is defined as Formula 14, where is subpopulation size and means combination.
(14) 
In this paper, Formula 14 is replaced by Formula 15 in order to decrease computation complexity.
(15) 
In Formula 15, is the best individual in subpopulation. represents each individual in it other than . For once computation of subpopulation diversity, this change makes that total times of the calculation for the difference between two individuals greatly reduce from to in each subpopulation. Although the average of distance between pairwise individuals is replaced by that from the best individual to another one may lead to some error in the resulting value of subpopulation diversity, convergence degree can still be reflected with much less computation. As the steps shown in Algorithm 2, of each subpopulation is computed at intervals to obtain each .
IvC Experiment to Compare Result
Our experiment is carried on a Drawing TC5000A computing platform. It has 1264 2.6 GHz cores. Its memory capacity is 1.5 TB. In our experiment, and are set different value combinations to find the fit relationship between and . In detail, their values are both get from the set, . In total, there are nine value combinations. Fig. 19 are graphs of the function expressed by Formula 4 under different value combination of and , respectively. Then, the outcomes under each value combination are compared with those of a traditional DEA, respectively. The algorithms in the experiment are both based on the EA introduced in Subsection 4.1 and are the same in setting of common parameters. The thinking of setting for these common parameters comes from [24]. In detail, common parameters except migration interval are set the same value in both algorithms. Then, for each instance, the traditional algorithm runs thirty times independently under five equal difference intervals, respectively. Under each of the nine value combinations of and , so does the DEA with our scheme. The value of common parameters except migration interval is listed in Table I.
Evolutionary parameters  Mutation rate ()  Changing during a run according to Formula 16 based on the initial value, 0.02. 

Crossover rate  1  
Mapping rate ()  Changing during a run according to Formula 17 based on the initial value, 0.05.  
Threshold value of evolutionary velocity  5000  
Parallel parameters  Migration topology  Ring 
Migration strategy  Randomrandom  
Migration size  1  
Subpopulation size  100  
Quantity of subpopulation  16  
Terminal criterion  2000 migration rounds having been done 
This table shows that mutation rate, , and mapping rate, , change during a run according to Formula 16 and Formula 17, respectively.
(16) 
(17) 
In Formula 16, is the initial . Similarly, is the initial in Formula 17. In both formulas, denotes current generations and represents the maximal ones.
Since the EA used in our experiment can find the optimum solution of many TSP instances smaller than a280, eight much larger instances in [15] from pcb442 to vm1084 are chosen in our experiment. The results of the traditional algorithm are listed in Table II. Those of the DEA based on our scheme under different value combinations of and are listed in Table IIIXI
, respectively. In these tables, each instance corresponds to five migration intervals. Under each interval, the average of outcomes and the standard deviation of them is given. Besides, the optimal solution of each instance provided by
[15] is listed in each table. In Table IIIXI, results having significant difference with those in Table IIin terms of ttest with 95% confidence are highlighted by
italics and bold.Instance  Interval 





pcb442  150000  50939.3  7.01  
200000  50937.7  7.84  
250000  50935.0  24.85  50778  
300000  50937.5  7.54  
350000  50934.6  10.94  
p654  50000  34643.4  1.22  
60000  34643.4  1.22  
70000  34643.5  1.57  34643  
80000  34643.0  0.00  
90000  34643.1  0.37  
d657  200000  49092.7  36.28  
250000  49084.5  27.21  
300000  49063.9  39.07  48912  
350000  49083.2  34.98  
400000  49065.3  35.79  
u724  150000  42143.3  43.23  
200000  42105.0  37.78  
250000  42086.1  43.66  41910  
300000  42085.6  32.77  
350000  42068.8  33.05  
rat783  150000  8829.1  11.71  
200000  8821.8  9.11  
250000  8818.9  7.82  8806  
300000  8815.6  5.97  
350000  8815.6  7.50  
dsj1000  800000  18786697.6  27651.24  
1000000  18756550.5  20746.29  
1200000  18761931.7  19044.51  18659688  
1400000  18762151.6  24886.79  
1600000  18757890.0  21787.65  
pr1002  500000  259680.9  253.01  
600000  259551.0  233.13  
700000  259553.3  249.21  259045  
800000  259399.3  159.45  
900000  259459.9  208.12  
vm1084  600000  239785.1  204.25  
800000  239773.2  177.70  
1000000  239779.5  187.50  239297  
1200000  239738.4  178.93  
1400000  239739.5  188.81 
Instance  Interval 




pcb442  150000  50936.8  8.37  
200000  50935.6  8.11  
250000  50927.8  24.62  50778  
300000  50924.8  33.10  
350000  50934.5  12.86  
p654  50000  34643.1  0.51  
60000  34643.1  0.37  
70000  34643.0  0.00  34643  
80000  34643.1  0.37  
90000  34643.1  0.37  
d657  200000  49077.9  32.29  
250000  49057.9  32.71  
300000  49059.6  31.23  48912  
350000  49057.0  30.24  
400000  49059.1  45.58  
u724  150000  42099.2  31.61  
200000  42074.6  33.18  
250000  42073.3  40.11  41910  
300000  42064.9  23.90  
350000  42060.3  36.93  
rat783  150000  8825.0  7.97  
200000  8817.3  7.62  
250000  8815.9  8.28  8806  
300000  8814.5  6.23  
350000  8814.8  5.52  
dsj1000  800000  18761927.0  25664.25  
1000000  18762398.9  23733.48  
1200000  18748623.4  24506.86  18659688  
1400000  18745199.9  27584.08  
1600000  18744510.3  23523.72  
pr1002  500000  259551.6  268.12  
600000  259504.9  223.44  
700000  259383.7  172.11  259045  
800000  259375.9  233.52  
900000  259387.2  239.40  
vm1084  600000  239736.0  164.15  
800000  239676.7  141.30  
1000000  239702.1  163.49  239297  
1200000  239682.9  179.91  
1400000  239637.6  122.91 
Instance  Interval 





pcb442  150000  50931.7  12.66  
200000  50934.3  10.46  
250000  50929.5  29.45  50778  
300000  50932.7  10.44  
350000  50927.1  22.43  
p654  50000  34643.0  0.00  
60000  34643.0  0.00  
70000  34643.0  0.00  34643  
80000  34643.0  0.00  
90000  34643.0  0.00  
d657  200000  49055.4  35.44  
250000  49057.3  42.35  
300000  49048.0  38.85  48912  
350000  49045.9  37.06  
400000  49036.6  32.54  
u724  150000  42070.7  33.66  
200000  42067.9  29.89  
250000  42054.6  31.99  41910  
300000  42037.9  39.68  
350000  42036.7  35.86  
rat783  150000  8822.8  9.91  
200000  8815.8  7.32  
250000  8816.2  7.01  8806  
300000  8812.9  5.72  
350000  8812.9  8.13  
dsj1000  800000  18756031.3  24623.43  
1000000  18748806.2  26546.89  
1200000  18748989.6  22840.60  18659688  
1400000  18735264.4  20742.59  
1600000  18738518.6  26541.26  
pr1002  500000  259433.6  190.88  
600000  259335.0  165.78  
700000  259303.5  140.85  259045  
800000  259280.7  138.09  
900000  259354.0  199.96  
vm1084  600000  239683.8  162.64  
800000  239665.7  136.93  
1000000  239671.2  135.21  239297  
1200000  239638.1  145.54  
1400000  239623.6  126.22 
Instance  Interval 





pcb442  150000  50932.6  11.09  
200000  50930.4  10.13  
250000  50926.6  27.04  50778  
300000  50919.6  32.26  
350000  50922.3  23.90  
p654  50000  34643.0  0.00  
60000  34643.0  0.00  
70000  34643.0  0.00  34643  
80000  34643.0  0.00  
90000  34643.0  0.00  
d657  200000  49050.0  31.05  
250000  49054.0  41.66  
300000  49045.9  34.37  48912  
350000  49028.7  31.89  
400000  49042.3  36.07  
u724  150000  42072.2  31.06  
200000  42045.2  27.53  
250000  42045.9  30.77  41910  
300000  42032.3  29.95  
350000  42026.7  26.53  
rat783  150000  8828.1  8.83  
200000  8817.9  6.69  
250000  8813.9  6.77  8806  
300000  8815.0  6.24  
350000  8813.4  6.44  
dsj1000  800000  18741740.2  22220.14  
1000000  18739983.0  24819.99  
1200000  18738972.7  18942.56  18659688  
1400000  18724773.5  27336.00  
1600000  18732647.5  20957.86  
pr1002  500000  259277.2  136.14  
600000  259402.0  201.64  
700000  259342.0  189.56  259045  
800000  259296.1  180.98  
900000  259288.9  132.95  
vm1084  600000  239691.4  169.62  
800000  239690.1  167.67  
1000000  239637.8  128.26  239297  
1200000  239638.7  171.76  
1400000  239585.6  133.70 
Instance  Interval 





pcb442  150000  50938.3  6.86  
200000  50937.4  8.53  
250000  50934.7  10.22  50778  
300000  50933.5  27.43  
350000  50921.9  35.34  
p654  50000  34643.3  1.18  
60000  34643.6  2.30  
70000  34643.1  0.37  34643  
80000  34643.0  0.00  
90000  34643.0  0.00  
d657  200000  49076.7  42.28  
250000  49083.5  33.96  
300000  49070.3  32.63  48912  
350000  49060.2  29.56  
400000  49058.7  42.51  
u724  150000  42093.8  39.47  
200000  42081.5  31.40  
250000  42081.8  30.52  41910  
300000  42064.9  28.86  
350000  42071.3  30.47  
rat783  150000  8825.9  10.50  
200000  8818.5  8.52  
250000  8816.4  6.64  8806  
300000  8814.4  6.72  
350000  8814.0  7.31  
dsj1000  800000  18773122.2  23162.37  
1000000  18761402.1  26171.98  
1200000  18755819.7  15957.70  18659688  
1400000  18746441.0  23247.20  
1600000  18752288.1  19447.99  
pr1002  500000  259517.0  273.21  
600000  259468.7  236.99  
700000  259449.1  199.82  259045  
800000  259515.5  272.15  
900000  259499.3  226.40  
vm1084  600000  239770.3  162.83  
800000  239730.6  137.43  
1000000  239693.5  151.11  239297  
1200000  239694.1  158.66  
1400000  239670.1  132.08 
Instance  Interval 





pcb442  150000  50931.5  20.76  
200000  50929.5  26.70  
250000  50933.9  12.78  50778  
300000  50931.7  10.29  
350000  50931.2  22.20  
p654  50000  34643.1  0.37  
60000  34643.1  0.37  
70000  34643.4  1.52  34643  
80000  34643.0  0.00  
90000  34643.0  0.00  
d657  200000  49067.4  43.30  
250000  49063.7  37.62  
300000  49055.5  36.67  48912  
350000  49050.3  32.32  
400000  49046.9  43.36  
u724  150000  42094.9  29.82  
200000  42064.9  42.94  
250000  42086.1  42067.8  34.59  
300000  42066.3  25.19  
350000  42054.6  30.69  
rat783  150000  8824.0  8.83  
200000  8819.7  7.47  
250000  8815.1  6.77  8806  
300000  8814.0  7.05  
350000  8812.5  6.69  
dsj1000  800000  18763625.5  21831.34  
1000000  18762785.8  24970.92  
1200000  18746381.2  20736.39  18659688  
1400000  18745791.7  17386.04  
1600000  18743302.0  18939.43  
pr1002  500000  259435.1  224.69  
600000  259417.3  222.38  
700000  259359.6  169.53  259045  
800000  259341.7  198.52  
900000  259317.7  187.90  
vm1084  600000  239739.4  183.33  
800000  239694.4  158.19  
1000000  239691.5  168.34  239297  
1200000  239699.0  143.17  
1400000  239668.0  129.28 
Instance  Interval 





pcb442  150000  50934.0  10.64  
200000  50931.3  11.15  
250000  50932.7  12.33  50778  
300000  50931.5  10.96  
350000  50927.6  14.41  
p654  50000  34643.0  0.00  
60000  34643.0  0.00  
70000  34643.0  0.00  34643  
80000  34643.1  0.37  
90000  34643.0  0.00  
d657  200000  49052.7  35.84  
250000  49053.1  36.05  
300000  49050.2  34.89  48912  
350000  49031.9  34.44  
400000  49041.6  47.01  
u724  150000  42070.6  37.78  
200000  42056.0  37.16  
250000  42035.9  27.50  41910  
300000  42031.8  30.09  
350000  42033.0  30.42  
rat783  150000  8824.6  9.78  
200000  8817.7  8.45  
250000  8814.6  6.62  8806  
300000  8812.7  6.50  
350000  8814.3  6.14  
dsj1000  800000  18740584.7  20775.90  
1000000  18743355.3  20831.26  
1200000  18741427.6  24245.91  18659688  
1400000  18729556.2  19250.64  
1600000  18733965.3  25656.26  
pr1002  500000  259394.7  204.43  
600000  259367.8  198.61  
700000  259331.4  180.84  259045  
800000  259410.4  179.61  
900000  259307.4  181.98  
vm1084  600000  239724.6  145.35  
800000  239717.7  149.77  
1000000  239638.4  148.45  239297  
1200000  239642.8  116.59  
1400000  239644.8  142.91 
Instance  Interval 





pcb442  150000  50933.8  9.27  
200000  50936.4  8.19  
250000  50934.8  10.53  50778  
300000  50936.1  9.49  
350000  50937.4  8.42  
p654  50000  34643.4  1.52  
60000  34643.1  0.37  
70000  34643.1  0.51  34643  
80000  34643.3  1.14  
90000  34643.0  0.00  
d657  200000  49085.9  38.80  
250000  49095.7  30.77  
300000  49075.7  29.14  48912  
350000  49065.2  40.71  
400000  49066.2  32.46  
u724  150000  42113.1  41.00  
200000  42097.8  41.37  
250000  42084.7  43.12  41910  
300000  42076.8  30.95  
350000  42077.5  37.93  
rat783  150000  8828.0  8.28  
200000  8818.5  6.06  
250000  8814.8  6.23  8806  
300000  8816.0  5.43  
350000  8814.6  5.70  
dsj1000  800000  18776618.6  27998.28  
1000000  18761359.1  26059.44  
1200000  18753840.4  22750.80  18659688  
1400000  18753787.1  26156.22  
1600000  18752323.7  21359.45  
pr1002  500000  259622.7  228.65  
600000  259523.0  314.84  
700000  259479.1  211.93  259045  
800000  259500.5  208.99  
900000  259519.0  196.58  
vm1084  600000  239765.1  122.05  
800000  239744.2  134.48  
1000000  239774.2  153.76  239297  
1200000  239675.7  164.24  
1400000  239661.6  125.02 
Instance  Interval 





pcb442  150000  50934.5  9.73  
200000  50935.6  25.76  
250000  50937.1  8.39  50778  
300000  50935.8  8.25  
350000  50935.2  11.55  
p654  50000  34643.3  0.69  
60000  34643.1  0.37  
70000  34643.0  0.00  34643  
80000  34643.0  0.00  
90000  34643.1  0.37  
d657  200000  49078.2  42.55  
250000  49078.7  47.63  
300000  49063.7  42.80  48912  
350000  49068.5  41.41  
400000  49065.7  40.75  
u724  150000  42112.0  30.64  
200000  42081.4  26.93  
250000  42080.1  38.89  41910  
300000  42069.0  38.67  
350000  42064.8  27.71  
rat783  150000  8827.5  8.08  
200000  8817.2  4.93  
250000  8816.7  8.42  8806  
300000  8816.5  6.18  
350000  8815.4  5.77  
dsj1000  800000  18762595.7  30773.90  
1000000  18765975.2  33663.58  
1200000  18759551.7  22376.08  18659688  
1400000  18755129.5  24348.98  
1600000  18753479.1  21177.11  
pr1002  500000  259586.6  235.77  
600000  259520.7  222.06  
700000  259454.7  244.08  259045  
800000  259351.2  196.07  
900000  259401.3  166.26  
vm1084  600000  239746.5  145.48  
800000  239690.4  160.28  
1000000  239742.5  204.31  239297  
1200000  239709.6  160.58  
1400000  239715.3  155.95 
Instance  Interval 




pcb442  150000  50937.6  8.36  
200000  50938.5  8.21  
250000  50935.4  10.23  50778  
300000  50934.7  10.39  
350000  50924.3  31.84  
p654  50000  34643.1  0.51  
60000  34643.0  0.00  
70000  34643.0  0.00  34643  
80000  34643.0  0.00  
90000  34643.0  0.00  
d657  200000  49060.4  42.18  
250000  49054.0  35.61  
300000  49046.5  39.59  48912  
350000  49050.9  30.57  
400000  49052.1  35.16  
u724  150000  42080.9  27.08  
200000  42089.8  35.98  
250000  42078.1  35.95  41910  
300000  42071.6  42.63  
350000  42052.2  39.33  
rat783  150000  8821.6  9.10  
200000  8819.5  8.56  
250000  8816.2  6.56  8806  
300000  8815.4  6.30  
350000  8814.8  5.12  
dsj1000  800000  18761976.8  25253.82  
1000000  18760042.3  14768.00  
1200000  18755479.9  22505.88  18659688  
1400000  18748354.6  22603.43  
1600000  18746204.2  23893.81  
pr1002  500000  259537.5  306.89  
600000  259412.2  244.67  
700000  259464.5  237.79  259045  
800000  259410.7  225.29  
900000  259372.2  182.52  
vm1084  600000  239789.1  252.07  
800000  239706.1  172.80  
1000000  239746.1  168.84  239297  
1200000  239653.7  146.65  
1400000  239660.9  140.98 
Based on the results in Table IIXI, the difficulty of a instance to the two algorithms, DF, can be expressed according to Formula 18, where denotes the average of all solutions for one instance obtained in our experiment and represents the optimal solution provided by [15].
(18) 
The value of difficulty is listed in Table XII.
Instance  pcb442  p654  d657  u724 
Difficulty  0.003048  0.000003  0.003040  0.003842 
Instance  rat783  dsj1000  pr1002  vm1084 
Difficulty  0.001337  0.004947  0.001482  0.001682 
According to Table XII
, the difficulty of each instance can be classified into three levels. That of p654 belongs to the lowest level. Rat783, pr1002 and vm1084 have an intermediate level difficulty. The rest four instances, pcb442, d657, u724, dsj1000, are ones with high difficulty.
It can be seen in Table IIXI that, for the lowest difficulty instance, p654, significant difference between solutions of the DEA based on our scheme and those of the traditional one can be found in no case. For the intermediate difficulty instances, the DEA with the proposed scheme significantly wins in fortyfive cases out of one hundred and thirtyfive ones (45/135) and statistically loses in two cases (2/135). It should be noted that, for high difficulty instances, the DEA with our scheme yields significantly better outcomes than its peer in ninety cases out of one hundred and eighty ones (90/180). Meanwhile, there are no significant differences in all the rest cases. Also, the tables show that the solutions’ standard deviation of the DEA with our scheme is less than that of the traditional one in two hundred and twentysix cases out of the all ones (226/360).
Moreover, it can been seen that, under different value combinations of and , the performance of the algorithm with our scheme is significant different. When , or , , for the all eight instances, the algorithm significantly wins in the most cases (27/40) and never statistically loses. When , , the DEA with our scheme also has a good behavior. In detail, it significantly wins in twentysix cases (26/40) and never statistically loses. On the whole, the winning rate is 135/360, while the losing rate is 2/360. Besides, under the three outstanding value combinations, solutions of the DEA with our scheme have a better standard deviation in eightytwo cases out of one hundred and twenty ones (82/120). This rate is better than that on the whole (226/360). Fig. 19 show that, under the three outstanding value combinations, the graph of the function in Formula 4 have the same characteristic which is distinguished from that under the other six ones.
In conclusion, results shows that the DEA based on our scheme has an advantage on solutions. Moreover, a value combination of and which makes the graphing slope of the function in Formula 4 increase monotonically from to zero when is fit for our scheme.
V Conclusion
In this paper, we have presented the scheme of setting the success rate of migration based on subpopulation diversity at each interval for DEAs. Under the control of the scheme, immigrants enter the target subpopulation at a probability, which is the function of subpopulation diversity according to Formula 4, at intervals. In our experiment, eight instances of the TSP are used to test algorithms. In detail, under nine different value combinations of parameters required for Formula 4, outcomes of the algorithm with the scheme are compared with those of the traditional DEA, respectively. The experimental results show that, especially for high difficulty instances, the DEA based on our scheme has a significant advantage on solutions. Moreover, under three value combinations of the parameters making graphing slope of the function in Formula 4 increase monotonically from to zero when , the algorithm with the scheme has most outstanding performance.
Based on the key factor in the behavior of evolutionary computation, diversity, we propose this scheme and discuss its parameters setting. This scheme can be used to improve solutions of DEAs for diversified problems. To apply the scheme, first of all, finding the diversity measure of a certain chromosome coding is necessary. In future, how to use this scheme together with some other diversity based methods should be studied to further improve DEAs. Also, the simplified calculation method for subpopulation diversity to reduce the time complexity used in our scheme should be discussed in theory.
Acknowledgment
The authors would like to thank Dr. Dunhui Xiao and assistant researcher Jian Wang for their valuable suggestions.
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