An improved artificial bee colony algorithm based on the strategy of global reconnaissance

Abstract

The artificial bee colony (ABC) algorithm is a recently introduced swarm intelligence optimization algorithm based on the foraging behavior of a honeybee colony. However, many problems are encountered in the ABC algorithm, such as premature convergence and low solution precision. Moreover, it can easily become stuck at local optima. The scout bees start to search for food sources randomly and then they share nectar information with other bees. Thus, this paper proposes a global reconnaissance foraging swarm optimization algorithm that mimics the intelligent foraging behavior of scouts in nature. First, under the new scouting search strategies, the scouts conduct global reconnaissance around the assigned subspace, which is effective to avoid premature convergence and local optima. Second, the scouts guide other bees to search in the neighborhood by applying heuristic information about global reconnaissance. The cooperation between the honeybees will contribute to the improvement of optimization performance and solution precision. Finally, the prediction and selection mechanism is adopted to further modify the search strategies of the employed bees and onlookers. Therefore, the search performance in the neighborhood of the local optimal solution is enhanced. The experimental results conducted on 52 typical test functions show that the proposed algorithm is more effective in avoiding premature convergence and improving solution precision compared with some other ABCs and several state-of-the-art algorithms. Moreover, this algorithm is suitable for optimizing high-dimensional space optimization problems, with very satisfactory outcomes.

Appendix. Step by Step Procedure of SABC Algorithm

Consider the optimization problem (dimension of the problem, $D=3$) as follows:

$$\begin{aligned} \mathrm{Minimize}\,f(x)=x_1^2 +x_2^2 +x_3^2 \quad {-}5\le x_1 ,x_2 ,x_3 \le 5 \end{aligned}$$

Control parameters of ABC algorithm are set as;

• Colony size, $m=7$

• The scale factor of scouts, alf = 0.1

• The number of scouts for global reconnaissance, $m_{s}=\hbox {round}(alf^{*}m)=1$ // round() towards nearest integer

• Number of food sources $(SN=(m-m_{s})/2=m_{e}=m_{o})$, $SN=3$

• Maximum number of trial for abandoning a source, $limit=200$

• Maximum number of fitness evaluations, $Max.FE=1000$

1. Initialization phase

num_FEs = 0

First, we initialize the positions of 3 food sources of employed bees randomly in the range [-5, 5] of uniform distribution.

$x=$

$$\begin{aligned} \begin{array}{rrr} -3.2064&{}\quad 1.8594&{}\quad 4.5068\\ 0.3521&{}\quad -1.3628&{}\quad 4.9516\\ -3.4553&{}\quad 2.1472&{}\quad 2.2129\\ \end{array} \end{aligned}$$

f(x) values are:

$$\begin{aligned}&34.0496\\&26.4996\\&21.4466 \end{aligned}$$

num_FEs = 3

Minimum objective function is 21.4466, the quality of the best food source.

Memorize best food source

$$\begin{aligned} \mathrm{Best} = \begin{array}{lll} -3.4553&{}\quad 2.1472&{}\quad 2.2129\\ \end{array} \end{aligned}$$

We initialize the global reconnaissance positions of a scout by Eq. (4)

$s=-1.0448\quad -1.3256\quad 4.8798$

f(s) value is: 26.6617

num_FEs = 4

$26.6617>21.4466$, the quality of the best food source could not be improved.

Memorize best food source

$$\begin{aligned} \mathrm{Best} = \begin{array}{lll} -3.4553&{} \quad 2.1472&{}\quad 2.2129\\ \end{array} \end{aligned}$$

2. Scouts global search phase

1st scout bee

• $r_{1}=0.1453$ //$r_{1}$ is a chaotic sequence by Eq. (6)

• $r_{2}=0.4066$ //$r_{2}$ is a chaotic sequence by Eq. (6)

• $j=1$ // j is a random selected dimension of the problem

• $s_{1}=0.6926\quad -1.3256\quad 4.8798$

• $f(s_{1})= 26.0496$

• num_FEs =5

• Apply greed selection between s and $s_{1}$

• $26.0496<26.6617$, the solution 1 was improved, then replace the solution s with $s_{1}$.

• $26.0496>21.4466$, the quality of the best food source could not be improved.

3. Employed bees search phase

1st employed bee

• $j=3$ // j is a random selected dimension of the problem

• $v_{11}=-3.2064\quad 1.8594\quad 2.8848$ // a new solution calculated by Eq. (2)

• $f(v_{11})=22.0608$

• num_FEs = 6

• Apply greed selection between $x_{1}$ and $v_{11}$

• $22.0608<34.0496$, the solution 1 was improved, replace the solution $x_{1}$ with $v_{11}$ and set its trial counter as 0.

• $v_{12}=-3.2064\quad 1.8594\quad 5.0000$ // a new solution calculated by Eq. (8)

• $f(v_{12})=38.7388$

• num_FEs=7

• Apply greed selection between $x_{1}$ and $v_{12}$

• $38.7388>22.0608$, the solution 1 could not be improved, increase its trial counter.

• $v_{13}=-3.2064\quad 1.8594\quad 1.0428$ // a new solution calculated by Eq. (9)

• $f(v_{13})= 14.8261$

• num_FEs = 8

• Apply greed selection between $x_{1}$ and $v_{13}$

• $14.8261<22.0608$, the solution 1 was improved, replace the solution $x_{1}$ with $v_{13}$ and set its trial counter as 0.

• $v_{14}=-3.2064\quad 1.8594\quad 1.5531$ // a new solution calculated by Eq. (10)

• $f(v_{14})= 16.1510$

• num_FEs = 9

• Apply greed selection between $x_{1}$ and $v_{14}$

• $16.1510>14.8261$, the solution 1 could not be improved, increase its trial counter.

• $v_{15}=-3.2064\quad 1.8594\quad 2.2129$ // a new solution calculated by Eq. (11)

• $f(v_{15})=18.6356$

• num_FEs = 10

• Apply greed selection between $x_{1}$ and $v_{15}$

• $18.6356>14.8261$, the solution 1 could not be improved, increase its trial counter.

2nd employed bee

• $j=2$ // j is a random selected dimension of the problem

• $v_{21}=0.3521\quad -0.4644\quad 4.9516$ // a new solution calculated by Eq. (2)

• $f(v_{21})= 24.8581$

• num_FEs = 11

• Apply greed selection between $x_{2}$ and $v_{21}$

• 24.8581 $<26.4996$, the solution 2 was improved, replace the solution $x_{2}$ with $v_{21}$ and set its trial counter as 0.

• $v_{22}=0.3521\quad -2.4914\quad 4.9516$ // a new solution calculated by Eq. (8)

• $f(v_{22})=30.8494$

• num_FEs = 12

• Apply greed selection between $x_{2}$ and $v_{22}$.

• $30.8494>24.8581$, the solution 2 could not be improved, increase its trial counter.

• $v_{23}=0.3521\quad 1.1591\quad 4.9516$ // a new solution calculated by Eq. (9)

• $f(v_{23})=25.9861$

• num_FEs = 13

• Apply greed selection between $x_{2}$ and $v_{23}$

• $25.9861>24.8581$, the solution 2 could not be improved, increase its trial counter.

• $v_{24}=0.3521\quad 4.2100\quad 4.9516$ // a new solution calculated by Eq. (10)

• $f(v_{24})=42.3665$

• num_FEs = 14

• Apply greed selection between $x_{2}$ and $v_{24}$

• $42.3665>24.8581$, the solution 2 could not be improved, increase its trial counter.

• $v_{25}=0.3521\quad 2.2338\quad 4.9516$ // a new solution calculated by Eq. (11)

• $f(v_{25})=29.6322$

• num_FEs = 15

• Apply greed selection between $x_{2}$ and $v_{25}$

• $29.6322>24.8581$, the solution 2 could not be improved, increase its trial counter.

3rd employed bee

• $j=3$ // j is a random selected dimension of the problem

• $v_{31}=-3.4553\quad 2.1472\quad 2.5693$ // a new solution calculated by Eq. (2)

• $f(v_{31})=23.1508$

• num_FEs = 16

• Apply greed selection between $x_{3}$ and $v_{31}$

• $23.1508>21.4466$, the solution 3 could not be improved, increase its trial counter.

• $v_{32}=-3.4553\quad 2.1472\quad 2.7248$ // a new solution calculated by Eq. (8)

• $f(v_{32})= 23.9743$

• num_FEs = 17

• Apply greed selection between $x_{3}$ and $v_{32}$.

• $23.9743>21.4466$, the solution 3 could not be improved, increase its trial counter.

• $v_{33}=-3.4553\quad 2.1472\quad 1.2439$ // a new solution calculated by Eq. (9)

• $f(v_{33})=18.0971$

• num_FEs = 18

• Apply greed selection between $x_{3}$ and $v_{33}$.

• $18.0971<21.4466$, the solution 3 was improved, replace the solution $x_{3}$ with $v_{33}$ and set its trial counter as 0.

• $v_{34}=-3.4553\quad 2.1472\quad 2.2129$ // a new solution calculated by Eq. (10)

• $f(v_{34})= 21.4466$

• num_FEs = 19

• Apply greed selection between $x_{3}$ and $v_{34}$.

• $21.4466>18.0971$, the solution 3 could not be improved, increase its trial counter.

• $v_{35}=-3.4553\quad 2.1472\quad 3.8022$ // a new solution calculated by Eq. (11)

• $f(v_{35})= 31.0065$

• num_FEs = 20

• Apply greed selection between $x_{3}$ and $v_{35}$.

• $31.0065>18.0971$, the solution 3 could not be improved, increase its trial counter.

  $$\begin{aligned} x= \begin{array}{rrr} -3.2064&{}\quad 1.8594&{}\quad 1.0428\\ 0.3521&{} \quad -0.4644&{}\quad 4.9516\\ -3.4553&{} \quad 2.1472&{}\quad 1.2439\\ \end{array} \end{aligned}$$
  $$\begin{aligned} f(x)=&14.8261\\&24.8581\\&18.0971 \end{aligned}$$

//Calculate the probability values p for the solutions x by means of their fitness values by Eq. (3)

$$\begin{aligned} p = 0.4097\\ 0.2508\\ 0.3395 \end{aligned}$$

4. Onlookers search phase

Produce new solutions $\upsilon _{i}$ for the onlookers from the solutions $x_{i}$ selected depending on pi and evaluate them.

1st onlooker bee

• $i=3$

• $v_{31}=-3.3624\quad 2.1472\quad 1.2439$ // a new solution calculated by Eq. (2)

• $f(v_{31})= 17.4634$

• num_FEs $=$ 21

• Apply greed selection between $x_{3}$ and $v_{31}$.

• $17.4634<18.0971$, the solution 3 was improved, replace the solution $x_{3}$ with $v_{31}$ and set its trial counter as 0.

• $v_{32}=0.2252\quad 2.1472\quad 1.2439$ // a new solution calculated by Eq. (8)

• $f(v_{32})=6.2084$

• num_FEs $=$ 22

• Apply greed selection between $x_{3}$ and $v_{32}$.

• $6.2084<17.4634$, the solution 3 was improved, replace the solution $x_{3}$ with $v_{32}$ and set its trial counter as 0.

• $v_{33}=-0.8802\quad 2.1472\quad 1.2439$ // a new solution calculated by Eq. (9)

• $f(v_{33})= 6.9323$

• num_FEs $=$ 23

• Apply greed selection between $x_{3}$ and $v_{33}$.

• $6.9323>6.2084$, the solution 3 could not be improved, increase its trial counter.

• $v_{34}=-3.4553\quad 2.1472\quad 1.2439$ // a new solution calculated by Eq. (10)

• $f(v_{34})= 18.0971$

• num_FEs $=$ 24

• Apply greed selection between $x_{3}$ and $v_{34}$

• $18.0971>6.2084$, the solution 3 could not be improved, increase its trial counter.

• $v_{35}=-5.0000\quad 2.1472\quad 1.2439$ // a new solution calculated by Eq. (11)

• $f(v_{35})= 31.1577$

• num_FEs $=$ 25

• Apply greed selection between $x_{3}$ and $v_{35}$

• $31.1577>6.2084$, the solution 3 could not be improved, increase its trial counter.

2nd onlooker bee

• $i=3$

• $v_{31}=0.2252\quad 2.1472\quad 0.1162$ // a new solution calculated by Eq. (2)

• $f(v_{31})= 4.6745$

• num_FEs $=$ 26

• Apply greed selection between $x_{3}$ and $v_{32}$.

• $4.6745<6.2084$, the solution 3 was improved, replace the solution $x_{3}$ with $v_{31}$ and set its trial counter as 0.

• $v_{32}=0.2252\quad 2.1472\quad 2.3055$ // a new solution calculated by Eq. (8)

• $f(v_{32})= 9.9763$

• num_FEs $=$ 27

• Apply greed selection between $x_{3}$ and $v_{32}$

• $9.9763>4.6745$, the solution 3 could not be improved, increase its trial counter.

• $v_{33}=0.2252\quad 2.1472\quad 1.0809$ // a new solution calculated by Eq. (9)

• $f(v_{33})= 5.8294$

• num_FEs $=$ 28

• Apply greed selection between $x_{3}$ and $v_{33}$

• $5.8294>4.6745$, the solution 3 could not be improved, increase its trial counter.

• $v_{34}=0.2252\quad 2.1472\quad 2.2613$ // a new solution calculated by Eq. (10)

• $f(v_{34})= 9.7744$

• num_FEs $=$ 29

• Apply greed selection between $x_{3}$ and $v_{34}$

• $9.7744>4.6745$, the solution 3 could not be improved, increase its trial counter.

• $v_{35}=0.2252\quad 2.1472\quad 2.2129$ // a new solution calculated by Eq. (11)

• $f(v_{35})=9.5579$

• num_FEs $=$ 30

• Apply greed selection between $x_{3}$ and $v_{35}$

• $9.5579>4.6745$, the solution 3 could not be improved, increase its trial counter.

3rd onlooker bee

• $i=1$

• $v_{11}=-0.0601\quad 1.8594\quad 1.0428$ // a new solution calculated by Eq. (2)

• $f(v_{11})= 4.5484$

• num_FEs $=$ 31

• Apply greed selection between $x_{1}$ and $v_{11}$.

• $4.5484<14.8261$, the solution 1 was improved, replace the solution $x_{1}$ with $v_{11}$ and set its trial counter as 0.

• $v_{12}=-1.4447\quad 1.8594\quad 1.0428$ // a new solution calculated by Eq. (8)

• $f(v_{12})= 6.6320$

• num_FEs $=$ 32

• Apply greed selection between $x_{1}$ and $v_{12}$.

• $6.6320>4.5484$, the solution 1 could not be improved, increase its trial counter.

• $v_{13}=-3.9699\quad 1.8594\quad 1.0428$ // a new solution calculated by Eq. (9)

• $f(v_{13})= 20.3051$

• num_FEs $=$ 33

• Apply greed selection between $x_{1}$ and $v_{13}$.

• $20.3051>4.5484$, the solution 1 could not be improved, increase its trial counter.

• $v_{14}=-3.4952\quad 1.8594\quad 1.0428$ // a new solution calculated by Eq. (10)

• $f(v_{14})=16.7611$

• num_FEs $=$ 34

• Apply greed selection between $x_{1}$ and $v_{14}$.

• $16.7611>4.5484$, the solution 1 could not be improved, increase its trial counter.

• $v_{15}=-3.4553\quad 1.8594\quad 1.0428$ // a new solution calculated by Eq. (11)

• $f(v_{15})= 16.4842$

• num_FEs $=$ 35

• Apply greed selection between $x_{1}$ and $v_{15}$.

• $16.4842>4.5484$, the solution 1 could not be improved, increase its trial counter.

  $$\begin{aligned} x= \begin{array}{rrr} -0.0601&{} \quad 1.8594&{} \quad 1.0428\\ 0.3521&{} \quad -0.4644&{} \quad 4.9516\\ 0.2252&{} \quad 2.1472&{} \quad 0.1162\\ \end{array} \end{aligned}$$
  $$\begin{aligned} f(x)=&4.6745\\&24.8581\\&4.5484 \end{aligned}$$

//Memorize best food source

$$\begin{aligned} \mathrm{Best} = -0.0601\quad 1.8594\quad 1.0428 \end{aligned}$$

5. Scout bee search phase

$$\begin{aligned} \mathrm{Bas} =&4\\&4\\&4 \end{aligned}$$

//There is no abandoned solution since limit = 200

//If the solution of which the trial counter value is higher than limit = 200, select a best solution obtained by the scouts to replace with the abandoned one.

$\mathrm{num}\_\mathrm{FEs}\,\,<\,\,\mathrm{Max.FE}$

The procedure is continued until the termination criterion is attained.