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Evolutionary Computation Methods for Search-Based Data Analytics Problems

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Research Article | Open Access

Volume 2021 |Article ID 1336929 | https://doi.org/10.1155/2021/1336929

Libin Hong, Chenjian Liu, Jiadong Cui, Fuchang Liu, "Mutation Strategy Based on Step Size and Survival Rate for Evolutionary Programming", Discrete Dynamics in Nature and Society, vol. 2021, Article ID 1336929, 13 pages, 2021. https://doi.org/10.1155/2021/1336929

Mutation Strategy Based on Step Size and Survival Rate for Evolutionary Programming

Academic Editor: Shi Cheng
Received29 Jul 2021
Accepted14 Sep 2021
Published15 Oct 2021

Abstract

Evolutionary programming (EP) uses a mutation as a unique operator. Gaussian, Cauchy, Lévy, and double exponential probability distributions and single-point mutation were nominated as mutation operators. Many mutation strategies have been proposed over the last two decades. The most recent EP variant was proposed using a step-size-based self-adaptive mutation operator. In SSEP, the mutation type with its parameters is selected based on the step size, which differs from generation to generation. Several principles for choosing proper parameters have been proposed; however, SSEP still has limitations and does not display outstanding performance on some benchmark functions. In this work, we proposed a novel mutation strategy based on both the “step size” and “survival rate” for EP (SSMSEP). SSMSEP-1 and SSMSEP-2 are two variants of SSMSEP, which use “survival rate” or “step size” separately. Our proposed method can select appropriate mutation operators and update parameters for mutation operators according to diverse landscapes during the evolutionary process. Compared with SSMSEP-1, SSMSEP-2, SSEP, and other EP variants, the SSMSEP demonstrates its robustness and stable performance on most benchmark functions tested.

1. Introduction

Evolutionary programming (EP) is a major evolutionary algorithm that attempts to find a global optimum for benchmark functions; mutation is the only available operator in EP. Cauchy [1], Gaussian [1], Lévy [2], and double exponential [3] distributions are used as mutation operators in EP, and the popular EP variants include fast evolutionary programming (FEP), classic evolutionary programming (CEP), and Lévy-based evolutionary programming (LEP).

The early version of EP employs a single-mutation operator to optimize functions. While researchers observed that EP employing a single-mutation operator has limitations, Kumar [4] proposed that different mutation operators can generate different step sizes on average; however, the same mutation operator with different parameter values can generate various step sizes. For example, the Cauchy distribution can generate larger random values, thereby allowing individuals to jump from local optima, unlike the Gaussian distribution. The Lévy distribution is compatible with two features by controlling the value of . Combining mutation operators was proposed in [4]; the mutation operators are classified into very small, small, medium, large, and very large mutation types. Researchers have proposed several mixed-mutation strategies for EP because different mutation operators have different characteristics. Improved FEP (IFEP) [5] uses both Gaussian and Cauchy distributions as mutations. Hong et al. [6] proposed a mixed-mutation strategy called evolutionary programming (MSEP), wherein the appropriate mutation operator is selected during evolution based on the probabilities of the four mutation operators. MSEP with a local fitness landscape (LMSEP) [7] is an upgraded mutation strategy in which a local fitness landscape is proposed based on MSEP. Ensemble strategies with adaptive EP [8] combine a population with mutation operators, allowing every population to benefit from the function calls. Hong et al. [9] proposed an EP variant that employs a mutation strategy which is step-size-based and has a self-adaptive mechanism (SSEP). In recent years, researchers have proposed the automatic design of mutation operators and adaptive mutation operators for EP using hyper-heuristic/genetic programming [1012].

The previously proposed different mutation strategies attempt to use different step size mutation operators during the evolutionary process; however, one of the issues is that mutation strategies may cause loss of step size control [13, 14]. We herein propose a novel mutation strategy that uses both “step size” and “survival rate” to control the selection of mutation operator/type for evolutionary programming (SSMSEP). Meanwhile, the standard deviation of the Gaussian distribution is updated if a Gaussian mutation is selected. In [9], principles were proposed to guide the mutation strategy. SSMSEP-1 and SSMSEP-2 are two variants of SSMSEP, which use either the survival rate or step size, respectively. In this study, the principles are optimized, and the experimental data illustrate that SSMSEP is more robust and stable than existing mixed-mutation strategies and single-mutation operators in most cases. The proposed mutation strategy conquers the loss of step size control on our tested benchmark functions.

Section 2 describes function optimization and the basic EP algorithm. The details of the mutation strategy of SSMSEP are introduced in Section 3. Section 4 describes the implementation of the proposed algorithm. In Section 5, experimental results are presented, and SSMSEP, SSMSEP-1, SSMSEP-2, SSEP, CEP, FEP, and LEP with different values are tested. A comparison among SSMSEP, SSMSEP-1, SSMSEP-2, SSEP, MSEP, and LMSEP is also presented in this section. We explain and discuss future work in Section 6. Section 7 summarizes and concludes the paper.

2. Function Optimization and Evolutionary Programming

The global minimization in can be formalized as a pair , where is a bounded set on and is an -dimensional real-valued function. The objective is to obtain a point such that is a global optimum on . More explicitly, it is necessary to obtain such that

Here, must be bounded and it does not need to be continuous or differentiable. The EP algorithm is described as follows [1, 15]:(1)Generate individuals for the initial population and set k = 1. Each individual is taken as a pair of real-valued vectors. (), . The strategy parameter as the initial value was set to 3.0.(2)Calculate the fitness value for each (), .(3)Each parent (), , generates a single offspring () (where ).The above two formulas are used to generate new offspring. The benchmark function is used to evaluate the fitness value. The factors and are set as and .(4)Calculate the fitness value for each offspring (), , according to .(5)Conduct pairwise comparison over the set of parents () and offspring (), . For each individual, Y opponents were chosen randomly from the set of parents and offspring with equal probability. The individual gets a “reward” if its fitness value is smaller than that of the opponents in the comparison.(6)Select the individuals out of both parents and their offspring, , which have the most rewards to be parents, for the next generation.(7)Stop when the end condition is met; otherwise, k++ and go to Step 3.

It is CEP [1], FEP [5], and LEP [2] when is a random number generated by Gaussian, Cauchy, and Lévy distributions, respectively. The above algorithm acts as a basic framework for SSMSEP, and the general parameter settings of EP are provided in Table 1.


ParameterSettings

in Table 2
100
10
1e − 3



300
300
300
300
300
300
300
30−12569.5
300
300
300
,

300
300
20.998
40.0003075
2−1.0316285
20.398
23
3−3.86
6−3.32
4−10.15
4−10.34
4−10.54

3. Step Size and Survival Rate-Based Mutation Strategy

Here, we introduce the strategy used to design a step size and survival rate-based mutation strategy (SSMSEP) and explain how it works. The motivation of the SSMSEP is to solve the drawback of SSEP [9]. In SSEP, one of the issues is that the evolutionary process may have insufficient usage of long step-size-based mutation operators in the earlier generation of EP, which is also called loss of step size control [13, 14]. A typical case is the optimization of ; the mean best value over 50 runs is much worse than that of most single-mutation operators. From the experimental results, we observed that SSEP may fall into local optima in earlier generations because of insufficient mutation with a mutation operator which can generate a long step size. To solve this problem, we introduce the “survival rate” to control the mutation types in the proposed mutation strategy. The number of surviving offspring will be recorded to evaluate whether a long step size mutation is sufficient. In the proposed strategy, “step size” and “survival rate” are two keys to control mutation selection and parameter updating. Compared with SSEP, there are two significant changes.(i)A “survival rate” is imported, and we work with “step size” to control mutation type selection.(ii)Parameter calculation and updating strategy for mutation operators are proposed in each generation of EP.

Liang et al. [13, 14] proposed the lower bound control of the offspring to improve the EP performance and pointed out that self-adaptation may swiftly cause a search step size that is too narrow to further scan the search space, which is called the loss of step size control. In addition, Liang et al. [13, 14] analyzed how step size control was lost. Yao et al. [5] indicated that the Gaussian mutation is more likely to produce offspring closer to its parent than a Cauchy mutation. Thus, a Gaussian mutation is a better selection if the individual is near the global optimum; in contrast, the Cauchy mutation is a better selection for EP. Hong et al. [9] designed an SSEP that can explore space with a long step size at the outset and afterwards use a short step size mutation operator in the search, where “step size” denotes the distance between a “survived” offspring and its parent. In this paper, we propose using both “step size” and “survival rate” to control the mutation operator with parameter updating.

EP uses a static mutation operator, which leads to a few offspring surviving in the later generations of the run. A mutation strategy that switches mutation operators with related parameters is necessary to improve the performance of EP. The idea is to design an algorithm that can search a wider space to guarantee that more of the search space can be explored at the outset, and that search for a narrower space can be conducted later. is a Gaussian distribution, and and represent the standard normal distribution. Usually, represents a mutation operator that can generate a short step size, and the Cauchy distribution represents a long step size mutation operator. In SSMSEP, is dynamically updated using the proposed equation. is set to 0.1, 0.01, or 1 in SSEP, where the step size cannot effectively prevent the EP loss of step size control on some benchmark functions. In SSMSEP, “step size” and “survival rate” are combined and calculated to evaluate whether a mutation with a long step size is sufficient.

3.1. Symbols Used in the Novel Mutation Strategy

The symbols used by the mutation strategy are as follows:(i) is a single real value; it represents the step size at generation . The new population comprises both “survived” parents and “survived” offspring after tournament selection at generation ; the parents and offspring which appear in the new population are called “survived” parents and offspring, respectively. is evaluated as the mean absolute value of the jumped step size of all surviving offspring. This value is updated at generation (in Algorithm 1 line 11), , where is the maximum EP generation.(ii) is taken as a single real-valued vector; it records the nonabsolute value of the step size of each individual i after tournament selection, , where is the population number; can be either a positive or a negative value.(iii) is a single real value; it represents the mean step size from generation 1 to generation . is evaluated as , where is the number of current generations. This value is recalculated in every generation as well (in Algorithm 1 line 13).(iv) is a single real value; it represents the survival rate at each generation . Each parent has an offspring after mutation, and after tournament selection, both parent and offspring have the opportunity to be selected as parents for the next generation. In SSMSEP, the number of selected offspring (which is summed up) divided by the population size is called the survival rate.(v) represents the Gaussian distribution; in SSMSEP, , and is updated dynamically.(vi) indicates the Cauchy distribution.(vii) is the distance coefficient; it helps EP to control the usage of the long step size mutation operator, when is very small, to prevent EP from being trapped in a local optimum prematurely. Table 3 provides the value of which is empirically determined for each benchmark function. The values of follow [9].(viii) represents survival rate; it is an empirically determined constant; it is set to 0.12.



150100150100150150150150150150150150150150150150150150150150150150150

(1) T/∗Set constant ∗/
(2) S_RATE/∗Set constant ∗/
(3) Update both popStepSize[1…POPNO] and popStepSizeRaw[1…POPNO] after tournament selection;
(4)  = popStepSizeRaw[1…POPNO];
(5)for i: POPNO do
(6)  if pop[i] is offspring then
(7)   totalPopStepSize + = popStepSize[i];
(8)   survivedOffspringNo ++;
(9)  end if
(10)end for
(11)  = totalPopStepSize/survivedOffspringNo;/∗Renew ∗/
(12)  = survivedOffspringNo/POPNO;/∗Renew survival rate∗/
(13)  = mean();/∗Renew ∗/
(14)ifthen
(15)  ;/∗Set mutation type C∗/
(16)else ifthen
(17)  ;/∗Set mutation type ∗/
(18)  ;/∗Renew for ∗/
(19)else
(20)  ;/∗Set mutation type ∗/
(21)  ;/∗Set 1 for ∗/
(22)end if
3.2. Mutation Strategy for SSMSEP

The step size and survival rate-based mutation strategy for EP is as follows:(i)If , , or survival rate and the mutation type is Cauchy distribution in current generation, then Cauchy distribution is selected for the next generation.(ii)If and and survival rate at the current generation, Gaussian distribution is selected and set to by the following equation, where is the population number, for the next generation:(iii)For all other cases, Gaussian distribution is selected and set and for , as the mutation operator for EP.

In this study, for the earlier generation of EP, the Cauchy distribution was used as the mutation operator. We used both the step size and survival rate to evaluate whether the Cauchy distribution was sufficiently applied. 0.12 is an empirically determined value based on a large number of experiments. When the step size is sufficiently small, a Gaussian distribution is applied with the updated for each generation. Instead of fixing the value of , which is set to 0.1 and 0.01, we control the shape of the Gaussian distribution in SSEP. The proposed strategy can also effectively avoid getting trapped in local optima during the evolutionary process.

4. SSMSEP Implementation

In this section, we present the pseudocode of the SSMSEP algorithm designed in accordance with the description of the mutation strategy in Section 3.2. The absolute value and non-absolute value of the step size for the population at each generation of EP are calculated using Algorithm 2. The calculation of and in Algorithm 1 uses the values prepared in Algorithm 2. Algorithm 1 implements the strategy proposed in Section 3.2 and describes how to switch the mutation operators with related parameters. Algorithms 1 and 2 are inserted into the EP algorithm described in Section 2.

(1)DIM/∗Set dimensional size∗/
(2)POPNO/∗Set population size∗/
(3)pop[POPNO 2][DIM]/∗Parents and offspring in present generation∗/
(4)for i: POPNO do
(5) indStepSize = 0;
(6)for j: DIM do
(7)  temp(i, j) = pop[i][j] - pop[POPNO + i][j];/∗Compute single step size for an individual∗/
(8)  indStepSize + = abs(temp(i, j));/∗Compute value of total absolute step size for overall individuals∗/
(9)  indStepSizeRaw + = temp(i, j);/∗Compute value of total step size for overall individuals∗/
(10)end for
(11) popStepSize(POPNO + i) = indStepSize/DIM;/∗Compute mean absolute step size for pop[i]∗/
(12) popStepSizeRaw(POPNO + i) = indStepSizeRaw/DIM;/∗Compute mean non-absolute step size for pop[i]∗/
(13)end for

To better observe the influence of “step size” and “survival rate” on the algorithm, we designed two variants of SSMSEP: SSMSEP-1 only uses “survival rate” ( in line 14, in line 16 in Algorithm 1) to select the mutation operator; SSMSEP-2 only uses “step size” ( in line 14, in line 16 in Algorithm 1) to select the mutation operator.

5. Experimental Results

Table 2 lists the 23 benchmark functions, which are also commonly used by other researchers [1, 5, 7, 9] in experiments. The benchmark functions include unimodal benchmark functions, multimodal benchmark functions with many local optima, and multimodal benchmark functions with a few local optima, specified as , , and , respectively [5]. The results of SSMSEP, SSMSEP-1, SSMSEP-2, SSEP, EP using Cauchy distribution (FEP), EP using Gaussian distribution (CEP), and EP using Lévy distributions with different values (1.2,1.4,1.6, and 1.8) are provided for each function in Table 4. We retain 4 digits after the decimal point on most benchmark functions in this table.


SSMSEPSSMSEP-1SSMSEP-2SSEPFEPCEP

Mean3.7814E − 081.0901E − 041.4066E − 075.5197E − 084.5635E − 046.0315E − 052.5421E − 041.6934E − 041.3440E − 041.0516E − 04
Min1.4472E − 082.5524E − 081.3595E − 087.0480E − 093.0863E − 043.2239E − 051.5908E − 041.2202E − 048.8775E − 056.3256E − 05
Std Dev(3.7792E − 08)(2.3378E − 04)(6.1386E − 07)(6.5450E − 08)(7.0235E − 05)(2.5166E − 05)(3.4971E − 05)(2.6149E − 05)(2.7323E − 05)(1.8110E − 05)

Mean3.6408E − 044.0167E − 043.2274E − 042.4660E − 047.0726E − 022.2304E − 025.1594E − 024.2239E − 023.6880E − 023.3471E − 02
Min2.9214E − 043.3439E − 042.6128E − 042.0121E − 045.1482E − 021.8824E − 024.3426E − 023.3697E − 023.2571E − 022.6264E − 02
Std Dev(3.0086E − 05)(3.0845E − 05)(2.8645E − 05)(1.5612E − 05)(5.5881E − 03)(1.5150E − 03)(3.2306E − 03)(3.0114E − 03)(2.1175E − 03)(2.5190E − 03)

Mean1.0407E − 024.8819E − 029.0411E − 039.3736E − 038.7540E − 031.0524E − 029.3985E − 037.1744E − 034.2293E − 033.0734E − 03
Min2.4395E − 038.1053E − 042.4140E − 032.3030E − 031.9117E − 038.5077E − 059.4563E − 047.5277E − 044.5609E − 042.2686E − 04
Std Dev(6.7484E − 03)(5.7250E − 02)(5.5701E − 03)(6.4085E − 03)(6.4514E − 03)(1.9002E − 02)(1.2368E − 02)(1.1998E − 02)(6.5552E − 03)(5.2651E − 03)

Mean6.4065E − 049.7131E − 018.8194E − 041.4405E − 036.7516E − 031.2210E + 006.5288E − 035.4754E − 022.8222E − 015.9637E − 01
Min3.1376E − 052.8928E − 032.8940E − 052.1072E − 055.4115E − 031.4146E − 014.2954E − 033.2552E − 033.6687E − 033.1680E − 03
Std Dev(1.8023E − 03)(8.1875E − 01)(1.4856E − 03)(4.0274E − 03)(7.8715E − 04)(9.0747E − 01)(5.6855E − 03)(1.4088E − 01)(4.8753E − 01)(6.0409E − 01)

Mean2.9275E + 013.15082.9490E + 012.1359E + 012.9350E + 017.54791.9830E + 016.38888.00591.1073E + 01
Min4.6981E − 029.9139E − 041.5319E − 035.5730E − 024.7199E − 011.8391E − 023.3474E − 011.0103E − 018.0644E − 024.2083E − 02
Std Dev(3.2710E + 01)(1.0064E + 01)(3.1230E + 01)(2.6718E + 01)(3.2291E + 01)(1.4348E + 01)(2.3933E + 01)(6.0507E + 00)(1.3649E + 01)(1.9445E + 01)

Mean064.98000136.14000.6229.88
Min0000000000
Std Dev(0)(128.02)(0)(0)(0)(367.07)(0)(0)(2.62)(183.78)

Mean8.4609E − 032.8136E − 028.1618E − 037.9013E − 038.3416E − 032.2382E − 028.4703E − 031.1287E − 021.2451E − 021.5274E − 02
Min2.7008E − 031.4340E − 024.0161E − 033.7373E − 033.7395E − 038.4446E − 034.7781E − 034.8277E − 035.7806E − 037.6249E − 03
Std Dev(2.6809E − 03)(9.8926E − 03)(2.1752E − 03)(2.1289E − 03)(2.8422E − 03)(7.6244E − 03)(2.4166E − 03)(3.6673E − 03)(3.9032E − 03)(5.3375E − 03)

Mean−1.1243E + 04−9.0460E + 03−1.1053E + 04−1.1113E + 04−1.1300E + 04−7.9096E + 03−1.0582E + 04−1.0070E + 04−9.4480E + 03−8.7650E + 03
Min−1.1977E + 04−1.1145E + 04−1.1740E + 04−1.1976E + 04−1.1977E + 04−9.0739E + 03−1.1661E + 04−1.0892E + 04−1.0870E + 04−1.0418E + 04
Std Dev(2.7196E + 02)(6.4104E + 02)(3.5576E + 02)(4.0896E + 02)(2.7570E + 02)(5.9723E + 02)(4.8018E + 02)(4.3464E + 02)(5.3691E + 02)(6.0867E + 02)

Mean3.3160E − 018.4770E + 013.8155E + 014.3687E + 013.7750E − 029.0465E + 012.9352E − 022.8450E + 001.9402E + 016.6639E + 01
Min2.9129E − 023.3829E + 013.1528E − 022.2605E − 022.3464E − 024.1792E + 011.8232E − 022.1729E − 025.9871E + 003.2844E + 01
Std Dev(1.0996E + 00)(2.4674E + 01)(2.3952E + 01)(2.9723E + 01)(8.6261E − 03)(2.3879E + 01)(1.3161E − 02)(1.7927E + 00)(6.9163E + 00)(2.1773E + 01)

Mean1.6879E − 031.9694E + 012.4643E − 038.7447E − 021.6157E − 028.1049E + 001.1862E − 021.1219E − 028.8736E − 014.4735E + 00
Min6.8753E − 056.7988E + 006.6043E − 054.7010E − 051.4046E − 023.4041E + 009.2344E − 037.6683E − 038.4415E − 031.1552E + 00
Std Dev(4.8030E − 03)(1.8609E + 00)(5.9462E − 03)(4.1917E − 01)(1.1968E − 03)(2.5300E + 00)(1.3762E − 03)(4.7517E − 03)(1.4152E + 00)(2.1722E + 00)

Mean2.2300E − 027.4742E − 022.2230E − 021.7921E − 022.4342E − 021.2093E − 014.1413E − 024.3552E − 026.7181E − 021.1395E − 01
Min1.4659E − 082.1056E − 099.3328E − 104.5294E − 101.5764E − 052.5166E − 069.3085E − 064.6435E − 065.5552E − 063.7370E − 06
Std Dev(2.7383E − 02)(5.9743E − 02)(3.4994E − 02)(1.9481E − 02)(2.8117E − 02)(1.6475E − 01)(4.3791E − 02)(7.0052E − 02)(7.6744E − 02)(2.2874E − 01)

Mean4.1633E − 031.3546E + 004.1507E − 034.1515E − 032.0812E − 031.1365E + 002.0741E − 021.5691E − 013.5859E − 014.4462E − 01
Min6.5400E − 102.0109E − 064.0296E − 101.5497E − 104.0330E − 068.8123E − 072.0292E − 061.8880E − 061.2414E − 069.6019E − 07
Std Dev(2.0519E − 02)(2.3403E + 00)(2.0521E − 02)(2.0520E − 02)(1.4661E − 02)(1.6131E + 00)(7.5513E − 02)(3.6762E − 01)(5.7222E − 01)(6.2836E − 01)

Mean3.9555E − 056.4014E + 006.8845E − 042.5325E − 048.5921E − 054.5179E + 007.1911E − 054.9155E − 035.8544E − 012.0819E + 00
Min6.3527E − 091.2170E − 062.2008E − 091.7797E − 095.6319E − 059.0896E − 063.2237E − 052.3006E − 051.9697E − 053.5239E − 05
Std Dev(5.0506E − 05)(6.9850E + 00)(2.6287E − 03)(1.5500E − 03)(1.8465E − 05)(5.9895E + 00)(4.6632E − 05)(1.4952E − 02)(1.7692E + 00)(3.2842E + 00)

Mean1.39461.75631.22051.39461.28051.63211.45421.53711.31771.7693
Min0.9980.9980.9980.9980.9980.9980.9980.9980.9980.998
Std Dev(8.7265E − 01)(1.1928E + 00)(5.0212E − 01)(8.7181E − 01)(5.7523E − 01)(1.1595E + 00)(8.9826E − 01)(9.8139E − 01)(6.7645E − 01)(1.4785E + 00)

Mean4.3569E − 044.5400E − 044.3569E − 045.0894E − 044.5400E − 044.0332E − 043.9906E − 045.0894E − 043.8074E − 044.9063E − 04
Min3.0749E − 043.0749E − 043.0749E − 043.0749E − 043.0749E − 043.0749E − 043.0749E − 043.0749E − 043.0749E − 043.0749E − 04
Std Dev(3.2096E − 04)(3.3910E − 04)(3.2096E − 04)(3.8317E − 04)(3.3910E − 04)(2.9089E − 04)(2.7749E − 04)(3.8317E − 04)(2.5094E − 04)(3.6999E − 04)

Mean−1.0316−1.0316−1.0316−1.0316−1.0316−1.0316−1.0316−1.0316−1.0316−1.0316
Min−1.0316−1.0316−1.0316−1.0316−1.0316−1.0316−1.0316−1.0316−1.0316−1.0316
Std Dev(1.0087E − 09)(3.1347E − 10)(3.9007E − 11)(1.7266E − 12)(3.0332E − 08)7.9887E − 09)(3.3852E − 08)(2.0453E − 08)(2.2805E − 08)(1.9422E − 08)

Mean0.39790.39790.39790.39790.39790.39790.39790.39790.39790.3979
Min0.39790.39790.39790.39790.39790.39790.39790.39790.39790.3979
Std Dev(1.2995E − 10)(3.0825E − 11)(2.2603E − 11)(7.8869E − 13)(7.0398E − 09)5.1135E − 09)(1.4545E − 08)(9.8416E − 09)(8.3266E − 09)(8.0297E − 09)

Mean3.00003.00003.00003.00003.00003.00003.00003.00003.00003.0000
Min3.00003.00003.00003.00003.00003.00003.00003.00003.00003.0000
Std Dev(3.5756E − 08)(4.3186E − 09)(2.8254E − 09)(1.0494E − 10)(1.1868E − 06)8.8630E − 07)(1.1530E − 06)(1.3493E − 06)(1.4221E − 06)(7.6097E − 07)

Mean−3.8628−3.8628−3.8628−3.8628−3.8628−3.8628−3.8628−3.8628−3.8628−3.8628
Min−3.8628−3.8628−3.8628−3.8628−3.8628−3.8628−3.8628−3.8628−3.8628−3.8628
Std Dev(2.6613E − 06)(2.0399E − 08)(2.0050E − 06)(1.3995E − 06)(1.9028E − 06)6.1612E − 07)(1.5250E − 06)(9.6434E − 07)(1.0641E − 06)(1.2872E − 06)

Mean−3.2842−3.2675−3.2563−3.2675−3.2437−3.2794−3.2675−3.2675−3.2651−3.2842
Min−3.3224−3.3224−3.3224−3.3224−3.3223−3.3224−3.3224−3.3224−3.3224−3.3224
Std Dev(5.9431E − 02)(6.0015E − 02)(5.9333E − 02)(6.0014E − 02)(5.7039E − 02)(5.6169E − 02)(6.0012E − 02)(6.0014E − 02)(6.0159E − 02)(5.6174E − 02)

Mean−6.4811−5.4657−6.5003−6.6111−6.4439−8.4928−6.3396−7.0739−7.3097−7.8652
Min−10.1532−10.1532−10.1532−10.1532−10.1531−10.1532−10.1532−10.1532−10.1532−10.1532
Std Dev(2.4491E + 00)(1.3963E + 00)(2.3030E + 00)(2.7105E + 00)(2.5162E + 00)(2.4704E + 00)(2.3080E + 00)(2.5335E + 00)(2.6909E + 00)(2.6294E + 00)

Mean−8.2317−5.8052−8.3806−8.3522−7.5690−9.6849−7.7189−8.1391−8.3426−8.5489
Min−10.4029−10.4029−10.4029−10.4029−10.4029−10.4029−10.4029−10.4029−10.4029−10.4029
Std Dev(2.7034E + 00)(2.2835E + 00)(2.8997E + 00)(2.8100E + 00)(2.9246E + 00)(1.9791E + 00)(3.0032E + 00)(2.8505E + 00)(2.8157E + 00)(2.7805E + 00)

Mean−8.7721−8.5261−8.3264−8.1230−8.5698−9.6378−8.7256−8.9864−8.5304−8.9635
Min−10.5364−10.5364−10.5364−10.5364−10.5364−10.5364−10.5364−10.5364−10.5364−10.5364
Std Dev(3.0564E + 00)(2.8798E + 00)(3.3848E + 00)(3.3768E + 00)(2.8178E + 00)(2.1817E + 00)(3.1400E + 00)(2.6928E + 00)(3.1457E + 00)(2.8776E + 00)

The results of the Wilcoxon signed-rank test within a 95% confidence interval among SSMSEP, SSMSEP-1, SSMSEP-2, SSEP, and other single-mutation operators are given in Table 5. “” and “” indicate that SSMSEP can perform better or worse on average, respectively, compared to a specific existing mutation strategy or mutation operator in this table; “” and “” denote statistical significance for better or worse on average, respectively.


NGsvs SSMSEP-1vs SSMSEP-2vs SSEPvs FEPvs CEPvs vs vs vs

1500
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100
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100
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100
100
100

5.1. Comparison and Analysis

The experimental data in Table 4 show that SSMSEP achieves the best performance on average compared with SSMSEP-1, SSMSEP-2, SSEP, and EP using a single-mutation operator on , , , , , and , the second-best performance on average of , and the third-best performance on average of , , and . The SSMSEP provides the best fitness value on , , and and the third-best fitness value on , , , , , and . , , , and have equivalent values for both the averaged values and best fitness values over 50 independent runs. and have equivalent best fitness values for all EP variants except for FEP, and and have equivalent best fitness values for all EP variants.

Table 5 lists the results of the 2-tailed Wilcoxon signed-rank tests, showing a statistically significant difference among SSMSEP and SSMSEP-1, SSMSEP-2, SSEP, CEP, FEP, and LEP variations. SSMSEP is significantly better than SSMSEP-1 on , , , , , , , , , , , , and ; it shows better performance on , , , and . SSMSEP is significantly better than SSMSEP-2 on , , and ; it shows better performance than SSMSEP-2 on , , , , , , and . SSMSEP is significantly better than SSEP on and ; it shows better performance than SSEP on , , , , , , , and .

Table 6 gives function evaluations (FEs) of SSMSEP, SSMSEP-1, SSMSEP-2, SSEP, CEP, FEP, and LEP with .