Elite leader Dwarf Mongoose Optimization Algorithm

doi:10.21203/rs.3.rs-4935575/v1

To improve the performance of original dwarf mongoose optimization algorithm, this study proposes an elite leader dwarf mongoose optimization algorithm (EL-DMOA). EL-DMOA adopts two stage structure. The leader stage employs differential operator to improve the selected leaders. The artificial fitness is introduced for selecting the swarm leaders. If one individual’s fitness ceasing to improve, the artificial fitness method will imposes additional punishment to the relevant individual, thereby the newly founded solution is encouraged to lead the swarm. In the follower stage, each individual learns from the leaders. The crossover operation is employed to enhance swarm diversity. The experiments on CEC2017 test suite and real-life application problems show that EL-DMOA performs better than FIPS, DE/rand/1 and four recently meta-heuristics.

Dwarf mongoose optimization algorithm

two level structure

artificial fitness

crossover

Inspired by nature phenomenon and animals behavior, the scholars proposed a lot of meta-heuristics. For example, Kennedy mimics the foraging behavior of bird swarm and proposed particle swarm optimization[1]. Seyedali imitates the wolf swarm’s e leadership and hunting mechanism and proposed grey wolf optimizer[2]. Learning from the honey badger’s foraging behavior, Fatma proposes Honey Badger Algorithm[3].

When coping with NP-hard problems, meta-heuristics can find the global optima or near optima within acceptable computing time. For meta-heuristics algorithms is not rely on mathematical model, they are quite convenient for coping with non-contineuous or non-differential problems. Due to the above mentioned merits, a lot of meta-heuristics are proposed each year. For example: Lai [4] transfers emotional learning into Parallel Social Spider Optimization Algorithm to overcome premature convergence. Gao [5] merges chaotic local search into JADE to avoid local optima. Seid [6] employs dual Simplex working together and proposes Stochastic Dual Simplex Algorithm(SDSA). Wang [7]enploys self-adaptive Gaussian scale factor and a variable scale factor to control the search behavior of spherical evolution algorithm.The test results show that SDSA performs better than PSO and a group of GWO algorithms on 25 well know benchmark functions. Neeraj [8] proposes Ananya Algorithm. Ananya Algorithm is designed in simple concept, it employs the difference between the mean variable and the best variable to generate new solution. To solve the urban transit routing problem, Vasileios [9] proposed an artificial fish swarm optimization algorithm (ASFO). Eugenio[10] proposes supervised parallel optimization framework to balance the exploration and exploitation of meta-heuristics. To improve global search capability for data classification, Lai [11] combines chimp optimization algorithm with tracking-learning and fuzzy opposition-learning. To deal with Nurse Rostering Problem, Zhang [12] proposes Multi-agent deep Q-network-based metaheuristic algorithm. Kamran [13] combines Clustering Wavelet Opposition-based with Marine Predators Algorithm, to enhance the ability of jumping out of local optima. To facilitate meta-heuristics adapt to diverse optimization problems, Jia [14]proposes a Memory backtracking strategy including thinking stage, recall stage, and memory stage.To realize evenly search and avoid local optima, Qian [15] proposes Mixed attraction and repulsion based gravitational search algorithm. To enhance exploration ability, Romio [16] transfers quasi-opposite numbers and chaotic-map into original Equilibrium Optimizer. To obtain the balance of exploration and exploitation, Karol [17] proposes a Attraction-Repul ion Optimization Algorithm. Tian [18] proposes a dual performance metrics-based mutation operator for differential evolution. Zhao [19] proposes Triangulation topology aggregation optimizer based on r triangular topology. Inspired by PID control theory, Gao [20] proposes PID-based search algorithm (PSA). The test results show that PSA can balance exploration and exploitation to obtain high performance on complex optimization problems.

Recently, Jeffrey imitates the dwarf mongoose’s foraging behavior proposes Dwarf Mongoose Optimization Algorithm(DMOA). Due to the high performance of DMOA, it arousing the interest of evolutionary research. Too further improve the performance of DMOA, this study adopts three strategy: (1) The elite individual acts as the leaders and gain an addition round of evolution to improve convergence. (2) Introduces artificial fitness for selecting swarm leader, the newly found excellent individuals are encouraged to lead the swarm. (3) Crossover is employed to preserve swarm diversity. The rest of this study is: Section Ⅱ reports the related works, Section Ⅲ is the methodologies, Section Ⅳ reports the experiments and Section Ⅴ concludes this study.

2.1 DMOA

The dwarf mongoose lives in the rocks, mounds, and hollow trees. The role of dwarf mongoose swarm can be divided into guards, babysitters, attacking predators according to their sex and age. The prey field of dwarf mongoose is unpredictable. The dwarf mongoose forging in a large space and avoid overlap visiting the same position. The whole swarm is divided into Alpha group, scout group and babysitters group. Each group employs different compensatory behavioral adaptation as below:

1. Alpha group: In the alpha group, the Alpha female is selected based on it’s fitness. The selection probability is calculated according to Eq. (1).

$$\:{\alpha\:}=\frac{{\text{f}\text{i}\text{t}}_{\text{i}}}{{\sum\:}_{\text{i}=1}^{\text{n}}{\text{f}\text{i}\text{t}}_{\text{i}}}$$

1

2. Alpha group generate the candidate food position according to Eq. (2).

$$\:{X}_{i+1}={X}_{i}+pℎi\ast\:peep$$

2

Phi∈[-1,1] is a uniform random number.

Scout group: The scout group generate new food position according to Eq. (3,4).

$$\:{X}_{i+1}=\left\{\begin{array}{c}{X}_{i}-CF\ast\:pℎi\ast\:rand\ast\:({X}_{i}-\overrightarrow{M})\:\:\:\:\:\:\:\:if\:{\phi\:}_{i+1}>{\phi\:}_{i}\\\:{X}_{i}+CF\ast\:pℎi\ast\:rand\ast\:({X}_{i}-\overrightarrow{M})\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:else\end{array}\right.$$

3

$$\:\text{C}\text{F}={(1-\frac{\text{i}\text{t}\text{e}\text{r}}{{\text{M}\text{a}\text{x}}_{\text{i}\text{t}\text{e}\text{r}}})}^{\frac{\text{i}\text{t}\text{e}\text{r}}{{\text{M}\text{a}\text{x}}_{\text{i}\text{t}\text{e}\text{r}}}}$$

4

rand∈[0,1] is a uniform random number. CF is employed to control the search step of Dwarf Mongoose. Iter and Max_iter denote for the index of iteration and the maximum iteration. $\:\overrightarrow{M}$ is the mean position of current swarm.

The babysitters subordinate to the young member, it works like re-initialize food position to replace the young member. Usually the size of babysitters is small. The babysitters helps the whole swarm to jump out local optimum according to Eq. (5).

$$\:{X}_{i+1}={L}_{M}+rand\ast\:({U}_{M}-{L}_{M})$$

5

3. The babysitters: The babysitters are subordinate to the group members. The babysitters reset the Unsatisfactory scouting and food source found by the group member to help them jump out of local optima.

2.2 important DMOA variants

Though DMOA has been proposed for only two years, it attracts widely attention and a few important variants are published. For example, Yazan [21] proposed Enhanced Dwarf Mongoose optimization algorithm based on deep-learning for drones attack detection. Zhang [22] proposed Combined Dwarf Mongoose Optimization (CDMO) algorithm to optimize Inception V4 deep-learning model. Bimal [23] employs Symbiotic Organism Search (SOS) to strengthen local search and proposed enhanced Dwarf Mongoose Optimization. Liu [24]proposes flexible Dwarf Mongoose Optimization for diagnose kidney stone. Hashim[25] transfers Alpha-Directed Learning Process into DMOA to boost searching ability. Souhil [26]employs Dwarf Mongoose Optimization for optimizing renewable energy power flow. Laila [27]proposes Quantum Dwarf Mongoose Optimization With Ensemble Deep Learning for intrusion detection.

3.1 Artificial Fitness

To encourage the impact of newly found elite solutions, the artificial fitness selection is proposed. The artificial fitness is based on the fitness of each individual. When one individual hasn’t been improved for a few iterations, is will be imposed a punishment according to Eq. (6,7). In other words, the recently updated individuals will gain high probability to be selected as leader.

$$\:Afit\left(i\right)=fit\left(i\right)+{Fit}_{dif}\ast\:(1-{e}^{(-{k}_{1}\ast\:stag(i\left)\right)})$$

6

$$\:{Fit}_{dif}=Fit\left(rank\right(0.7\left)\right)-Fit\left(rank\right(0.3\left)\right)$$

7

A _fit denote for artificial fitness. Afit is based on the fitness while modifies with the state of the individual’s state. Fit(rank(0.7)) denote for the individual’s fitness with the top 70% rank. Hence this equation is to avoid employing the abnormal punishment to the stagnated individual. With artificial fitness, the recently founded excellent solution is encouraged to guide the motion of the current population.

3.2 Leader stage

Though the Roulette Wheel Selection can enhance the impact of good individuals, it may lead a few individual updates for several times in one iteration. This is termed as over-evolved. For other individuals aren’t evolved sufficiently, they can’t provide valuable information for the over-evolved individuals and causing waste of evolution resource.

Algorithm 1 The pseudo code of the leader stage
1	Ser = randperm(nPop)
2	For j = 1:nLeader
3	[~,idx] = min(aFit(Ser(3j−2)), aFit(Ser(3j−1)), aFit(Ser(3*j)));
4	Leader(j) = idx(3*j−3 + idx);
5	Newpos = Pos( Leader(j),:) + 0.5*(Pos( r1,:)-Pos(r2,:))
6	Execute 90% probability crossover
7	Update Pos(Leader(i),:) by Newpos according to greedy selection
8	End

To avoid over-evolved, this study adopt tournament selection to select leaders (Ref. Algorithm.1). For example, selecting the best individual from three random selected individuals, and the used individuals will not attend the next round of tournament selection in this iteration. 90% probability crossover denote for update 90% dimensions with new position and keep 10% dimensions the same as previous iteration. The greedy selection mean if the new generated solution is better than the original solution, the original solution is updated by new solution. nPop and nLeader denote for the number of population and the number of leaders.

3.3 follower stage

In the follower stage, every individual is updated in each iteration. Each follower learns from one leader selected from leader poor (Ref. Algorithm2). For the elite leaders gain additional round of evolution, they can speed up the convergence rate of swarm. Stag(i) records the number of iteration of the individual’s fitness ceasing to improve. Stag(i) is employed to calculate artificial for selection of leaders.

Algorithm 2 The pseudo code of the follower stage
1	For i = 1:nPop
2	iLeader = Leader(ceil(rand*Nleader)) % Select a leader from the leader pool
3	NewPos = Pos( r1,:) + 0.5*(Pos( iLeader,:)-Pos(r2,:))
4	Execute 90% crossover
5	Update Pos(i,:) by Newpos according to greedy selection
6	If Pos(i,:) is updated
7	Stag(i) = 0
8	Else
9	Stag(i) = Stag(i) + 1
10	End
11	End

3.4 The Dwarf Mongoose stage

In this stage, the Dwarf Mongoose motion is employ to explore the whole search space. The position of each individual is updated as Eq. (8,9)

$$\:Pos(i,:)=\left\{\begin{array}{c}Pos(i,:)-CF\ast\:pℎi\ast\:rand\ast\:\left(Pos\right(i,:)-\overrightarrow{M})\:\:\:\:\:\:\:\:if\:{\phi\:}_{i+1}>{\phi\:}_{i}\\\:Pos(i,:)+CF\ast\:pℎi\ast\:rand\ast\:\left(Pos\right(i,:)-\overrightarrow{M})\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:else\end{array}\right.$$

8

$$\:\text{C}\text{F}={(1-\frac{\text{i}\text{t}\text{e}\text{r}}{{\text{M}\text{a}\text{x}}_{\text{i}\text{t}\text{e}\text{r}}})}^{\frac{\text{i}\text{t}\text{e}\text{r}}{{\text{M}\text{a}\text{x}}_{\text{i}\text{t}\text{e}\text{r}}}}$$

9

rand∈[0,1] is a uniform random number. CF is employed to control the search step of Dwarf Mongoose. Iter and Max_iter denote for the index of iteration and the maximum iteration. $\:\overrightarrow{M}$ is the mean position of current swarm.

3.5 The proposed method

The process of the proposed methods is show in Fig. 1.

In the initialization, the population and major parameters are initialized.

Then the individuals’ artificial fitness are calculated and the leaders are selected.

The leaders are evolved according to algorithm 1. Only a few excellent individuals are selected and evolved in this step. Any leader can only be evolved for one time in each iteration.

Evolve the population according to algorithm 2.

Update the population’s position according to Dwarf Mongoose’s motion, to search the optimization sufficiently.

If the iteration number reaches the maximum iteration number, output the results. Else return to the next iteration.

To validate the effectiveness of the proposed EL-DMOA, thirty CEC2017test suite[28] and seven real-life application problems are employed. The experiment results is compared with original DMOA, one classic PSO, one classic DE and three meta-heuristics. For EL-DMOA, the swarm size and number of leaders are set as 100 and 20, respectively. All the parameters setting of comparison algorithms adopt the authors suggested values.

4.1 benchmark test

The test results in Table 1 indicates that EL-DMOA yield high performance, it generate the lowest mean errors on 21 functions. EL-DMOA obtains the best performance on all three unimodal functions and eight functions out of ten hybrid functions. DE/rand/1 and DMOA yield the lowest mean errors on six and four functions, respectively. The original algorithm of DMOA is relatively better than AOA, PSA and TTAO. The wilcoxon-signed rank test results in Table 2 show that EL-DMOA outperforms other peer algorithms on over twenty functions. On composition functions, EL-DMOA outperforms all comparison algorithm with relatively weak advantages. Comparing with original DMOA, EL-DMOA wins on twenty four functions, ties on two functions and loses on four function. With elite leader to guide the evolution process, EL-DMOA performs much better than DMOA. The average rank of mean errors of FIPS, DE/rand/1, AOA, PSA, TTAO, DMOA and EL-DMOA are 4.23, 2.77, 6.60, 4.27, 3.6, 5 amd 1.5, respectively. The average rank of EL-DMOA is significant lower than any other peer algorithm. In other word, EL-DMOA yields overall better performance them other comparison algorithms on CEC2017 test suite.

Table 1

Mean performance on 30D CEC2017 functions
	FIPS	DE/rand/1	AOA	PSA	TTAO	DMOA	EL-DMOA
f₁	(4.17 ± 4.26)E + 03	(1.42 ± 0.18)E-14	(3.17 ± 12.7)E + 08	(3.75 ± 4.47)E + 03	(2.48 ± 2.91)E + 03	(5.60 ± 9.59)E + 03	(0.00 ± 0.00)E + 00
f₂	(3.48 ± 10.9)E + 14	(2.86 ± 14.5)E + 06	(6.16 ± 33.7)E + 30	(7.87 ± 16.4)E + 12	(1.03 ± 3.48)E + 02	(1.84 ± 4.41)E + 38	(0.00 ± 0.00)E + 00
f₃	(4.25 ± 1.71)E + 03	(3.71 ± 3.64)E + 01	(2.81 ± 2.61)E + 03	(9.67 ± 0.00)E + 02	(1.86 ± 2.47)E-01	(2.51 ± 5.46)E + 05	(1.70 ± 4.14)E-05
f₄	(1.22 ± 0.24)E + 02	(5.86 ± 0.00)E + 01	(9.42 ± 2.91E + 02	(6.07 ± 2.23)E + 01	(7.77 ± 3.13)E + 01	(3.07 ± 1.73)E + 01	(5.20 ± 2.09)E + 01
f₅	(1.37 ± 0.14)E + 02	(1.76 ± 0.11)E + 02	(2.06 ± 0.26)E + 02	(1.16 ± 0.44)E + 02	(1.22 ± 0.24)E + 02	(1.93 ± 0.10)E + 02	(3.11 ± 1.18)E + 01
f₆	(2.87 ± 0.19)E-08	(3.92 ± 4.15)E-08	(4.40 ± 1.22)E + 01	(2.80 ± 10.6)E-01	(1.86 ± 0.71)E + 01	(4.93 ± 6.34)E-13	(4.06 ± 11.3)E-07
f₇	(1.98 ± 0.10)E + 02	(2.14 ± 0.09)E + 02	(4.83 ± 1.25)E + 02	(1.91 ± 0.77)E + 02	(1.10 ± 0.24)E + 02	(2.23 ± 0.12)E + 02	(6.14 ± 1.38)E + 01
f₈	(1.44 ± 0.15)E + 02	(1.79 ± 0.11)E + 02	(1.33 ± 1.95)E + 02	(1.17 ± 0.34)E + 02	(1.04 ± 0.20)E + 02	(1.92 ± 0.13)E + 02	(2.61 ± 8.96)E + 01
f₉	(5.68 ± 5.78)E-14	(0.00 ± 0.00)E + 00	(3.31 ± 0.84)E + 03	(2.72 ± 1.14)E + 03	(3.46 ± 4.60)E + 02	(1.89 ± 4.31)E-14	(8.77 ± 17.8)E-02
f₁₀	(6.24 ± 0.35)E + 03	(6.74 ± 0.34)E + 03	(3.54 ± 0.64)E + 03	(3.30 ± 0.90)E + 03	(3.55 ± 0.37)E + 03	(7.74 ± 0.30)E + 03	(2.28 ± 0.88)E + 03
f₁₁	(7.86 ± 2.17)E + 01	(5.65 ± 1.50)E + 01	(1.31 ± 0.73)E + 02	(1.39 ± 0.59)E + 02	(1.01 ± 0.29)E + 02	(8.12 ± 0.78)E + 01	(1.48 ± 1.84)E + 01
f₁₂	(4.85 ± 3.60)E + 05	(6.30 ± 4.96)E + 03	(9.93 ± 33.0)E + 06	(4.54 ± 21.2)E + 04	(3.48 ± 1.75)E + 04	(2.00 ± 1.12)E + 07	(1.05 ± 0.90)E + 04
f₁₃	(1.86 ± 1.76)E + 04	(8.15 ± 0.63)E + 01	(7.00 ± 55.6)E + 04	(1.04 ± 1.12)E + 04	(7.72 ± 8.48)E + 03	(8.31 ± 6.64)E + 03	(3.04 ± 1.60)E + 01
f₁₄	(5.87 ± 3.33)E + 03	(6.34 ± 0.63)E + 01	(1.15 ± 0.84)E + 04	(1.06 ± 4.96)E + 04	(2.13 ± 1.82)E + 03	(4.23 ± 2.06)E + 04	(1.99 ± 1.11)E + 01
f₁₅	(1.34 ± 0.99)E + 04	(3.85 ± 0.57)E + 01	(6.43 ± 6.12)E + 03	(5.83 ± 0.12)E + 03	(2.65 ± 3.59)E + 03	(1.64 ± 1.89)E + 03	(6.54 ± 3.58)E + 00
f₁₆	(7.64 ± 2.97)E + 02	(6.79 ± 4.08)E + 02	(1.27 ± 0.30)E + 03	(1.09 ± 0.28)E + 03	(9.41 ± 1.92)E + 02	(1.77 ± 0.16)E + 03	(6.64 ± 3.29)E + 02
f₁₇	(1.62 ± 0.54)E + 02	(7.37 ± 0.70)E + 01	(7.52 ± 2.59)E + 02	(5.69 ± 2.29)E + 02	(2.52 ± 1.02)E + 02	(7.48 ± 1.22)E + 02	(6.36 ± 6.98)E + 01
f₁₈	(3.34 ± 1.70)E + 05	(3.75 ± 0.31)E + 01	(1.76 ± 1.50)E + 05	(1.16 ± 0.34)E + 05	(9.96 ± 7.13)E + 04	(4.46 ± 1.94)E + 06	(2.30 ± 0.97)E + 01
f₁₉	(7.63 ± 7.75)E + 03	(1.90 ± 0.61)E + 01	(174 ± 3.06)E + 04	(1.03 ± 0.21)E + 04	(4.33 ± 4.01)E + 03	(2.10 ± 2.52)E + 03	(5.18 ± 1.87)E + 00
f₂₀	(1.87 ± 0.82)E + 02	(3.01 ± 1.83)E + 01	(3.67 ± 15.9)E + 02	(5.75 ± 2.03)E + 02	(3.40 ± 0.78)E + 02	(7.08 ± 1.67)E + 02	(1.28 ± 1.30)E + 02
f₂₁	(3.32 ± 1.95)E + 02	(3.69 ± 0.01)E + 02	(3.73 ± 0.27)E + 02	(3.12 ± 0.44)E + 02	(2.91 ± 0.34)E + 02	(3.98 ± 0.13)E + 02	(2.28 ± 0.11)E + 02
f₂₂	(4.93 ± 15.0)E + 02	(1.00 ± 0.00)E + 02	(3.12 ± 1.72)E + 03	(2.51 ± 2.20)E + 03	(5.15 ± 12.6)E + 02	(7.89 ± 0.26)E + 03	(1.00 ± 0.00)E + 02
f₂₃	(4.59 ± 0.17)E + 02	(5.21 ± 0.10)E + 02	(8.83 ± 2.62)E + 02	(4.89 ± 1.08)E + 02	(4.69 ± 0.26)E + 02	(5.45 ± 0.08)E + 02	(3.77 ± 0.11)E + 02
f₂₄	(5.68 ± 0.20)E + 02	(5.89 ± 0.09)E + 02	(6.99 ± 1.99)E + 02	(5.68 ± 1.50)E + 02	(5.25 ± 0.30)E + 02	(6.27 ± 0.11)E + 02	(4.49 ± 0.14)E + 02
f₂₅	(3.91 ± 0.02)E + 02	(3.87 ± 0.00)E + 02	(4.27 ± 0.25)E + 02	(4.08 ± 24.5)E + 02	(4.08 ± 0.22)E + 02	(3.79 ± 0.00)E + 02	(3.87 ± 0.00)E + 02
f₂₆	(1.70 ± 0.58)E + 03	(2.38 ± 0.41)E + 03	(2.08 ± 1.73)E + 03	(2.15 ± 2.04)E + 03	(2.30 ± 0.77)E + 03	(2.64 ± 0.09)E + 03	(1.13 ± 0.20)E + 03
f₂₇	(5.49 ± 0.08)E + 02	(4.88 ± 0.07)E + 02	(5.08 ± 0.49)E + 02	(5.43 ± 1.40)E + 02	(5.41 ± 0.18)E + 02	(5.00 ± 0.00)E + 02	(4.99 ± 0.09)E + 02
f₂₈	(4.09 ± 0.08)E + 02	(3.21 ± 0.43)E + 02	(4.96 ± 0.25)E + 02	(3.68 ± 0.54)E + 02	(4.00 ± 0.30)E + 02	(5.00 ± 0.02)E + 02	(3.44 ± 0.61)E + 02
f₂₉	(6.99 ± 0.89)E + 02	(5.83 ± 1.32)E + 02	(1.04 ± 0.39)E + 03	(8.86 ± 2.47)E + 02	(8.99 ± 1.58)E + 02	(1.37 ± 0.26)E + 03	(4.31 ± 0.40)E + 02
f₃₀	(3.56 ± 2.84)E + 04	(2.00 ± 0.04)E + 03	(7.00 ± 31.2)E + 04	(5.48 ± 22.0)E + 03	(3.97 ± 1.13)E + 03	(5.80 ± 4.94)E + 02	(2.02 ± 0.06)E + 03
Best	0	6	0	0	0	4	21
Note:The format of data in the bracket is (Mean errror ± standard deviation), Best denote for the times of best performance achieved by the relevant algorithm. The best results is highlighted in bold.

Table 2

Wilcoxon signed-rank test results
	FIPS	DE/rand/1	AOA	PSA	TTAO	DMOA	EL-DMOA
f₁	1.21E-12/5	2.09E-09/2	1.21E-12/7	1.21E-12/4	1.21E-12/3	1.21E-12/6	1
f₂	1.21E-12/5	3.45E-07/3	1.21E-12/6	4.57E-12/4	4.56E-12/2	1.21E-12/7	1
f₃	3.02E-11/6	3.02E-11/3	3.02E-11/5	3.79E-01/4	3.02E-11/2	3.02E-11/7	1
f₄	1.58E-11/6	3.29E-04/3	2.15E-10/7	2.19E-04/4	2.54E-08/5	1.62E-04/1	2
f₅	3.02E-11/4	3.02E-11/5	3.02E-11/7	4.08E-11/2	3.02E-11/3	3.02E-11/6	1
f₆	6.60E-01/2	5.86E-01/3	2.03E-11/7	2.03E-11/5	2.03E-11/6	8.91E-02/1	4
f₇	3.02E-11/4	3.02E-11/5	3.02E-11/7	3.02E-11/3	6.12E-10/2	3.02E-11/6	1
f₈	3.01E-11/4	3.01E-11/5	3.01E-11/7	3.01E-11/3	3.01E-11/2	3.01E-11/6	1
f₉	4.47E-01/3	2.78E-03/1	9.33E-12/7	9.33E-12/6	9.33E-12/5	1.85E-01/2	4
f₁₀	3.02E-11/5	3.02E-11/6	8.84E-07/4/	1.09E-05/2	9.06E-08/3	3.02E-11/7	1
f₁₁	2.83E-10/3	5.03E-08/2	4.89E-11/7	4.43E-11/6	7.27E-11/5	5.97E-11/4	1
f₁₂	3.02E-11/5	9.05E-02/1	3.02E-11/6	1.29E-09/4	3.65E-08/3	3.02E-11/7	2
f₁₃	3.02E-11/6	3.02E-11/2	3.02E-11/7	3.02E-11/5	3.02E-11/3	3.02E-11/4	1
f₁₄	3.02E-11/4	3.02E-11/2	3.02E-11/6	3.02E-11/5	3.02E-11/3	3.02E-11/7	1
f₁₅	3.02E-11/7	3.02E-11/2	3.02E-11/6	3.02E-11/5	3.02E-11/4	3.02E-11/3	1
f₁₆	3.26E-01/3	7.28E-01/2	3.35E-08/7	2.28E-05/5	4.98E-04/4	3.34E-11/6	1
f₁₇	5.53E-08/3	4.64E-05/2	3.69E-11/7	4.98E-11/5	3.82E-09/4	3.02E-11/6	1
f₁₈	3.02E-11/6	1.21E-10/2	3.02E-11/5	3.02E-11/4	3.02E-11/3	3.02E-11/7	1
f₁₉	3.02E-11/5	4.50E-11/2	3.02E-11/7	3.02E-11/6	3.02E-11/4	3.02E-11/3	1
f₂₀	1.76E-02/3	2.15E-02/1	7.04E-07/7	3.16E-10/5	5.09E-08/4	4.08E-11/6	2
f₂₁	3.02E-11/4	3.02E-11/5	3.02E-11/7	3.02E-11/3	5.57E-10/2	3.02E-11/6	1
f₂₂	3.69E-11/3	3.62E-12/1	1.72E-12/6/	4.21E-12/5	1.94E-11/4	1.72E-12/7	1
f₂₃	3.02E-11/2	3.02E-11/5	3.02E-11/7	3.02E-11/4	3.02E-11/3	3.02E-11/6	1
f₂₄	3.02E-11/3	3.02E-11/5	3.02E-11/7	3.02E-11/4	9.92E-11/2	3.02E-11/6	1
f₂₅	3.02E-11/4	2.91E-09/2	5.57E-10/7	8.48E-09/5	1.11E-06/6	3.02E-11/1	3
f₂₆	3.57E-06/2	5.07E-10/5	7.06E-01/7	1.68E-03/3	9.06E-08/4	3.02E-11/6	1
f₂₇	3.02E-11/6	8.88E-06/1	2.90E-01/7	3.02E-11/5	3.02E-11/4	1.86E-01/3	2
f₂₈	1.16E-04/5	2.46E-02/1	3.58E-11/7	1.56E-03/3	2.71E-03/4	1.42E-11/6	2
f₂₉	3.02E-11/3	3.82E-09/2	4.08E-11/7	3.02E-11/4	3.02E-11/5	3.02E-11/6	1
f₃₀	3.02E-11/6	2.71E-01/2	2.71E-02/7	3.02E-11/5	3.02E-11/4	5.57E-10/1	3
	26/4/0	23/4/3	28/2/0	29/1/0	28/0/2	24/2/4
	4.233333333	2.766666667	6.6	4.266666667	3.6	5	1.5

4.2 Convergence test

To compare the convergence characteristics of El-DMOA, the convergence figures of four different types functions are presented in Fig. 2–5. On unimodal f₁, EL-DMOA converge fast and achieves zero error. DE ranks the second. EL-DMOA performs much better than original DMOA. On simple multimodal f₆, EL-DMOA converges fast at the beginning, while it is staged in the middle stage. DMOA FIPS and DE surpasses EL-DMOA in the middle stage. On hybrid function f₁₆, EL-DMOA takes the lead in the early stage. DE yields almost the same mean errors while DE converges relatively slowly. On composition function f₂₆ EL-PSO outperforms other algorithms. The convergence figures show that EL-DMOA can employ the elite individuals’ information effectively to improve search efficiency. EL-DMOA converge fast and can achieve high performance on most of the tested functions.

4.3 Real-life test

To evaluate the application potential of EL-DMOA, a group of Process Synthesis problems [29] are employed. The summary of test problems is given in Table 3

Table 3

Summary of Process Synthesis problems
Problem	D	g	h
RW8	2	2	0
RW9	3	1	1
RW10	3	3	0
RW11	7	4	4
RW12	7	9	0
RW13	5	3	0
RW14	10	10	0
Note: D stand for number of dimension. g and h denote for numbers of inequality constraint and equality constraint, respectively.

In Table 4 MF denotes for the mean fitness of 30 independent runs, MV stands for the mean violation of feasible objective function. FR and SR stand for the feasible solution rate and successful solution rate. The results in Table 4 show that EL-DMOA breed generates the best performance on RW9, RW10, RW11, RW12 and RW13. On RW8, the performance of EL-DMOA is quite close to the best performance. On RW11, the FR of EL-DMOA is higher than other peer algorithms. Besides EL-DMOA, DE/rand/1 and TTAO obtain relatively high performance.

Table 4

Test results on Process Synthesis problems
		FIPS	DE/rand/1	AOA	PSA	TTAO	DMOA	EL-DMOA
RW8	MF	2.05E + 00	2.00E + 00	1.96E + 00	2.00E + 00	2.00E + 00	1.95E + 00	2.00E + 00
	MV	4.62E-01	1.48E-17	1.70E-02	1.26E-16	0.00E + 00	2.48E-01	8.88E-17
	FR	16.7%	100%	63.33%	100%	100%	46.67%	100%
	SR	0.00%	100%	63.33%	100%	100%	0.00%	100%
RW9	MF	2.22E + 00	2.56E + 00	2.56E + 00	2.64E + 00	2.95E + 00	2.60E + 00	2.56E + 00
	MV	9.14E-01	0.00E + 00	4.18E-05	0.00E + 00	0.00E + 00	3.07E-01	0.00E + 00
	FR	0.00%	100%	100%	100%	100%	3.33%	100%
	SR	0.00%	100%	90.00%	83.33%	3.33%	3.33%	100%
RW10	MF	6.67E-01	1.08E + 00	1.16E + 00	1.08E + 00	1.08E + 00	1.10E + 00	1.08E + 00
	MV	5.97E-01	0.00E + 00	8.61E-02	0.00E + 00	0.00E + 00	2.14E-01	0.00E + 00
	FR	0.00%	100%	93.33%	100%	100%	3.33%	100%
	SR	0.00%	100%	50.00%	96.67%	100%	3.33%	100%
RW11	MF	1.75E + 02	1.61E + 02	1.34E + 02	1.31E + 02	1.29E + 02	1.26E + 02	1.07E + 02
	MV	3.50E + 01	1.00E + 00	3.44E + 01	7.00E-01	5.42E-01	6.59E + 00	2.33E-01
	FR	0.00%	0%	0.00%	30.00%	6.67%	13.33%	76.67%
	SR	0.00%	0%	0.00%	0.00%	0.00%	0.00%	0.00%
RW12	MF	3.42E + 4	2.92E + 00	4.49E + 00	3.02E + 00	2.99E + 00	5.97E + 4	2.92E + 00
	MV	1.92E + 04	0.00E + 00	1.56E + 00	0.00E + 00	0.00E + 00	4.70E + 03	0.00E + 00
	FR	0.00%	100%	100%	100%	100%	0.00%	100%
	SR	0.00%	100%	16.67%	16.67%	6.67%	0.00%	100%
RW13	MF	2.78E + 04	2.69E + 04	2.69E + 04	2.69E + 04	2.69E + 04	2.70E + 04	2.69E + 04
	MV	5.12E + 00	0.00E + 00	4.22E-01	0.00E + 00	0.00E + 00	9.37E-01	0.00E + 00
	FR	3.33%	100%	100%	100%	100%	16.67%	100%
	SR	0.00%	0%	0.00%	0.00%	0.00%	0.00%	0.00%
RW14	MF	1.10E + 05	5.85E + 04	5.83E + 04	5.99E + 04	5.51E + 04	7.07E + 04	5.85E + 04
	MV	4.25E + 03	0.00E + 00	4.68E + 03	0.00E + 00	1.43E-14	6.76E + 02	0.00E + 00
	FR	0.00%	100%	100%	100%	100%	30.00%	100%
	SR	0.00%	0%	0.00%	0.00%	0.00%	0.00%	0.00%

This paper proposes two staged structured elite leader dwarf mongoose optimization algorithm (EL-DMOA). In the leader stage, the differential operator is utilized to evolve the elite leaders. The artificial fitness is introduced to encourage the influence of new founded solutions. In the follower stage, the individuals learns for elite leaders. The experiments on CEC2017 test suite indicate that EL-DMOA performs much better than FIPS, DE/rand/1, AOA, AOA, PSA, TTAO and DMOA covering both classical algorithms and recently algorithm. The convergence test tells that EL-DMOA can employ the elite leaders information to obtain high search efficiency. In the real-life application test, EL-DMOA yields high performance on seven Process Synthesis problems. The test results indicates that EL-DMOA is application to complex real-life problems.

Acknowledgement:

The author thanks the anonymous reviewers for their constructive comments. The Applied Characteristic Disciplines of Electronic Science and Technology of Xiangnan University (XNXY20221210), the Scientific Research Start-up Fund for High-level Talents in Xiangnan University, Chenzhou Low Carbon Intelligent Manufacturing Technology Research and Development Center.

Competing Interests：

The authors declare no conflict of interest on financial or personal relationship.

Ethical and informed consent for data used

All the authors ensured that the data meets legal requirements and consented for data using.

Data availability:

Data provided within the manuscript is openly available.

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No competing interests reported.

Elite leader Dwarf Mongoose Optimization Algorithm

Status:

Version 1

Abstract

Figures

1 Introduction

2 Related works

2.1 DMOA

2.2 important DMOA variants

3 Methodologies

3.1 Artificial Fitness

3.2 Leader stage

3.3 follower stage

3.4 The Dwarf Mongoose stage

3.5 The proposed method

4 Experiment

4.1 benchmark test

4.2 Convergence test

4.3 Real-life test

5 Conclusions

Declarations

References

Additional Declarations

Status:

Version 1