PSO-based improved multi-flocks migrating birds optimization (IMFMBO) algorithm for solution of discrete problems

Soft Computing (2019) 23:5469–5484 https://doi.org/10.1007/s00500-018-3199-5

M E T H O D O L O G I E S A N D A P P L I C A T I O N

PSO-based improved multi-flocks migrating birds optimization

(IMFMBO) algorithm for solution of discrete problems

Vahit Tongur1 _{· Erkan Ülker}2 Published online: 18 April 2018

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract

In this paper, we proposed an improved migrating birds optimization algorithm to solve discrete problem. It is a metaheuristic search algorithm that is inspired by V formation during the migration of migratory birds. Proposed algorithm has two main modifications on basic migrating birds algorithm. Firstly, multi-flocks are used instead of single flock in order to avoid local minimum. Secondly, these flocks interact with each other for the more detailed search around flock that has got better solutions. This interaction is inspired by particle swarm optimization algorithm. Also, insertion method is used for neighborhood in migrating birds optimization algorithm. As a discrete problem, traveling salesman problem is chosen. Performance of the proposed algorithm is tested on some of symmetric benchmark problems from TSPLIB. Obtained results show that proposed method is superior to basic migrating birds algorithm.

Keywords Migrating birds optimization· Traveling salesman problem · Particle swarm optimization · Multi-flocks

1 Introduction

Discrete optimization problems are easy to define, but their mathematical expressions and solutions are difficult. In dis-crete problems, there are disdis-crete parameters that have limited values instead of continuous parameters that have infinite values.

There are many discrete optimization problems that can represent real-life problems nowadays. In particular, prob-lems in manufacturing, scheduling, transportation and health areas have attracted the attention of researchers. Traveling salesman problem (TSP) is well known of them. In this prob-lem, a salesman visits all cities only one time and returns to starting point (Zhong et al.2007; Ma et al.2008). The pur-pose of this problem is to find the shortest path for a tour

Communicated by V. Loia.

B

eulker@selcuk.edu.tr

1 _{Department of Computer Engineering, Necmettin Erbakan}

University, Konya, Turkey

2 _{Department of Computer Engineering, Selcuk University,}

Konya, Turkey

of a salesman. In this paper, TSP is selected and solved as a discrete problem.

Let G = (N, A) be a scalar graph; N = {1, 2, 3, . . . , n} demonstrates cities and A= N x N shows the paths between cities. In this case, the distance matrix between all city pairs can be expressed by D= (di j)nxn. Also, the permutation of a solution can be expressed asπ = {π1, π2, . . . , πn}. In TSP,

the objective function can be defined as in Eq.1, since the objective is the minimum travel distance.

min f(π) = n−1  i=1

d_πiπi+1+ dπnπ1 (1)

Although solution of TSP seems easy, its computational complexity increases exponentially with number of cities. Therefore, TSP is a NP-hard combinatorial optimization problem and it cannot be solved with conventional methods within reasonable time.

Conventional methods purpose optimal solutions while metaheuristic methods present approximate solutions. In spite of this, especially in large-scale problems, required solution time exceed the reasonable boundaries. There-fore, metaheuristic methods prefer instead of conventional methods for solving these problems due to increasing com-putational complexity.

Researchers carry on to develop new metaheuristic meth-ods in order to solve the NP-hard optimization problems. Up to now, lots of metaheuristic methods have applied to a lot of problems and high quality results are obtained. The biggest advantages of these methods are flexibility and easy-applicability. However, some algorithms produce better solutions than others in significant problems so that there is not a single metaheuristic method for all problems.

Many of metaheuristic algorithms have been applied to solve TSP for decades, such as Hybrid Discrete Artificial Bee Colony (HDABC) (Marinakis et al.2016), Dimensional Ant Colony Optimization (DACO) (Dutta et al.2015), Lin-Kernighan (LK) (Helsgaun 2000; Karapetyan and Gutin

2011), Simulated Annealing (SA) (Wang et al.2009; Meer

2007; Geng et al.2011), Genetic Algorithm (GA) (Liu and Zeng2009), Particle Swarm Optimization (PSO) (Shi et al.

2007; Zhong et al.2007), Improved Ant Colony Optimiza-tion (IACO) (Tuba and Jovanovic2013), Discrete Invasive Weed Optimization (DIWO) (Zhou et al.2015), Tabu Search (TS) (Zhong et al. 2008), Discrete Artificial Bee Colony Algorithm (DABC) (Kıran et al.2013), Improved Fruit Fly Optimization Algorithm (IFOA) (Huang et al. 2017) and Hybrid Differential Evolution Algorithm (HDE) (Wang and Xu2011).

Migrating Birds Optimization (MBO) algorithm that is a novel metaheuristic algorithm introduced by Duman et al. (2012) and the MBO is used to calculate minimum cost in quadratic assignment problems (QAP). In their study, per-formance of the MBO is compared perper-formance of DE, PSO, GESA, TS, SS and GA meteheuristic algorithms and the MBO is offered for QAP problems. Duman and Elikucuk (2013) used the MBO to solve credit card fraud detection problem. Niroomand et al. (2015) suggested the MBO in order to solve closed loop layout problems with modification in neighborhood structure. This modification is explained as regenerating of generated neighbor solu-tions with using mutation and crossover operators. Pan and Dong (2014) used an improved migrating birds optimiza-tion algorithm on hybrid flow shop scheduling with total flow time minimization problem. Initial population is pro-duced by mixed GRASP-NEH methods instead of randomly generating. Also, insertion and pairwise exchange methods are used simultaneously. Tongur and Ülker (2014) imple-mented the MBO to flow shop sequencing problem. In another work of authors, seven different neighbor struc-ture of MBO are examined on the TSP Tongur and Ülker (2016). Gao et al. (2013) offered the enhanced migrat-ing birds optimization algorithm in order to solve no-wait flow shop scheduling problem. In their method, initial population is generated by the help of NEH, SDH and EDY heuristic algorithms. In addition to these, insertion, swap and double-job-insert neighborhood structures are used.

It can be seen from the literature search, some parts of metaheuristic algorithms are changed in order to obtain better results while metaheuristic algorithms are applied to problems. This alteration is usually realized on local search methods. In the literature, it has been seen that the inser-tion method is more successful than the swap method for solving the TSP problem (Kıran et al. 2013; Tongur and Ülker2016). Therefore, the insertion method is determined as neighborhood structure of the MBO algorithm in this paper. However, only the alteration of neighborhood structure may not increase the quality of results at satisfactory level in solu-tion of TSP with some algorithms. Therefore, multi-flocks are used instead of single flock in order to avoide from local optimums and investigating search space more detailed. And these flocks are provided interaction with each other like communication particles at the PSO algorithm. The PSO algorithm is inspired by the observations of bird and fish flocks and every individual in the population is influenced by the global best solution. In the proposed algorithm, each flock is simulated as a particle in the PSO algorithm and thus the interaction between the flocks are provided.

Rest of the paper is organized as follows. The basic MBO and the PSO are introduced in Sects.2and3, respectively. The IMFMBO algorithm and apply to TSP is explained in Sect. 4. Results that obtained from experiments are tested and proposed method compared with some metaheuristic algorithms, which tested on the benchmark problems from TSPLIB, and the basic MBO in Sect. 5. Lastly, concludes with discussion in Sect.6.

2 Migrating birds optimization algorithm

MBO algorithm, which introduced by Duman et al., is inspired from V-shaped flying position in order to energy sav-ing when migratsav-ing of migratory birds (Duman et al.2012). The main reason of V-shaped sequencing during flying is interpreted as reducing of energy for flying of rearward bird thanks to turbulence that produce with flapping of front bird. During this trip, leader of the birds is the most energy con-sumer. Rest of the birds except leader, spend less energy by the help of air turbulence that produced by in front of them. When the leader is tired, other bird, which follow the leader, substitute leader and tired leader goes to end of flock and birds carry on flying with new sequence. MBO algorithm, which inspired from this behavior of migratory birds, is started with randomly generated solutions, which each of solution repre-sent a bird, from search space. Any solution (generally first generated solution) from this population is selected as leader solution and other solutions are placed right and left sides of leader solution. In this way, V-shaped migratory bird flock is created. MBO aim to develop current solutions in every step after created of initial population. Neighborhood structure

PSO-based improved multi-flocks migrating birds optimization (IMFMBO) algorithm for solution... 5471

Fig. 1 Neighborhood and sharing for k= 3 and x = 1

Fig. 2 Replacement of the leader

is used like lots of metaheuristic algorithm for local search. Insertion and pairwise exchange are local search methods which commonly used in binary coded discrete problems. However, success of these methods can be change depend-ing on the problem.

A lot of metaheuristic algorithm, which used the flock intelligence, communicate with special methods in popula-tion. Individuals in the PSO algorithm (Poli et al.2007) flock communicate with velocity equation while ants in the ACO algorithm (Dorigo et al.2006) communicate with following to pheromone tracks. MBO provides communication of birds in the flock with special method, which special for birds, and it is called as profit mechanism. This process in MBO is started with neighbor solution that provided by leader solu-tion. Number of neighbor solutions, which is produced for leader and other solutions, are determined to Eqs.2and4, respectively.

k∈ N+; k ≥ 3; k = 3, 5, 7, . . . (2) x∈ N+; 1 ≤ x ≤ (k − 1)/2 (3)

n= k − x (4)

where k represents the number of neighbor solution for the leader bird, x represents the number of shared neighbor solution and n represents the number of neighbor solution except leader bird.

As shown in Fig.1, leader solution generates three(k = 3) neighbor solutions. After that these produced neighbor

solu-tions are sorted to determine according to objective function from best to worst. The best neighbor solution is used to improve its solution. Rest neighbor solutions are shared with other solutions that follow the leader solution. Number of shared neighbor solution (x) is determined according to Eq. 3. According to this x = 1 is determined for k = 3. Firstly, sharing is started with left side that following the leader and second neighbor solution of leader solution is shared. Then, third neighbor solution of the leader solution is shared to solution on the right side that follows the leader solution. Except leader solutions, n solutions are generated as indicated in Eq.4due to this sharing. As in leader solution, the best solution among the generated neighbor solutions are used to improve its solution and x solutions are shared to other following solution. After this sharing, rest neighbor solutions are discarded. These neighbor solution generating and neighbor sharing process carry on up to end of the flock. These process is given in Fig.1.

Flock is flown at the same sequence until reach the prede-termined m value (number of flap) and then leader solution go back of the flock for rest. The following solution is assigned as new leader of the flock. The first replacement of the leader is started on the left side of the flock and carry on, respec-tively, left and right. Replacement of the leader solution is given in Fig.2.

When analyzed the general structure of the MBO, the change of leader solution is directed the entire flock toward the best of the current solution. Each solution is reinforced

local search by creating a neighboring solution within itself. The sharing mechanism helps to improve weak solutions. Despite these advantages, the MBO algorithm can be easily down to local optima. A new approach has been proposed to overcome these disadvantages.

Algorithm of the MBO is given in Algorithm 1 for mini-mization problems.

p = number of initial solutions.

k = number of neighbor solutions for leader solution. n = number of neighbor solutions for remaining (except leader

solution) solutions.

x = number of shared neighbor solutions with the following

solution.

m = number of flap (tour)

Generate p initial solutions randomly and place them on V formation (S1, S2,. . ., Sp)

Choose one of them as leader

Build two list from remaining p− 1 solutions as left (SL1, SL2,. . .

,SL(p−1)/2) and right (SR1, SR2,. . . ,SR(p−1)/2)

le f t si de= true

i= 0

while i< (problemsize)3do for j= 0 to m do

Generate k neighbor solutions of the leader and sort according to objective function from best to worst (L B1, L B2, . . . , L Bk)

i= i + k

if f(L B1) < f (leader) then

leader= L B1

end if

add L B2, L B4, . . . , L B2x to the neighbor solution set of SL1

and add L B3, L B5, . . . , L B2x+1to the neighbor set of SR1

t= 1

while t< (p − 1)/2 do

Generate k− x neighbor solutions of SLt

(N L1, N L2, . . . , N Lk) and SRt(N R1, N R2, . . . , N Rk)

and sort according to objective function from best to worst

i= i + 2(k − x) if f(N L1) < f (SLt) then SLt= N L1 end if if f(N R₁) < f (SRt) then SRt= N R1 end if

add N L2, N L3, . . . , N Lx+1to the neighbor set of SL(t+1)

add R L2, RL3, . . . , RLx+1to the neighbor set of SR(t+1) t= t + 1

end while end for

if le f t si de= true then

Move the leader to the end of left list and assign SL1as the

new leader else

Move the leader to the end of right list and assign SR1as the

new leader end if

le f t si de=!lef tside

end while

return the best solution in the flock

Algorithm 1: MBO algorithm

3 Particle swarm optimization (PSO)

algorithm

Particle swarm optimization (PSO) is a population-based metaheuristic optimization technique proposed by Eberhart and Kennedy (1995). The PSO algorithm is inspired by the observations of bird and fish flocks. These flocks are often exhibited behavior synchronize influenced by each other in search of food.

PSO is a population-based metaheuristic algorithm, and each solution represents an individual in the solution space. In the PSO algorithm, these individuals are called particles. Each particle has a fitness value and a speed vector that con-trols its movement in the flock. The next position of a particle is influenced by its speed.

One of the basic steps of this algorithm is to create the initial population. The PSO algorithm is started with particles that have random positions and velocities. At the next steps, the position of each particle is updated depend on pBestand

gBestvalues:

pBest is the best solution of a particle found until now.

gBest is the best solution that found in flock until now.

Letv and pos be the velocity and position of a particle, respectively. The new velocity and position of the particle are updated according to Eqs.5and6, respectively.

vt+1_{= wv}t_{+ c}

1r1(pBest− post)

+ c2r2(gBest− post) (5)

post+1= post+ vt+1 (6)

where c1and c2are the learning factors and generally take

the value of 2. r1 and r2 are randomly distributed random

values in the range [0, 1]. Thew value is called the inertia factor. Thew parameter is reduced at each iteration to reduce the previous speed effect.

4 PSO-based improved multi-flocks

migrating birds optimization (IMFMBO)

algorithm

Many metaheuristic algorithms can easily down to local optima. To avoid this, the algorithm used is often modi-fied.These alterations are usually in the initial populations and neighborhood structure. In addition, a hybrid algorithm can be created by taking the strengths of some metaheuristic algorithms. Poorzahedy and Rouhani devised seven differ-ent hybrid structures using metaheuristic algorithms such as genetic algorithm, simulated annealing and tabu search, and tested on the Sioux Falls network (Poorzahedy and Rouhani

PSO-based improved multi-flocks migrating birds optimization (IMFMBO) algorithm for solution... 5473

Fig. 3 Multi-flocks

2007). Debels et al. (2006) presented a hybrid algorithm that is combined scatter search and electromagnetism metaheuris-tic to solve scheduling problem. Jolai et al. (2012) combined population-based simulated annealing, adapted imperialist competitive algorithm and hybridisation of adapted imperi-alist competitive algorithm and population-based simulated annealing to solve the flow shop scheduling problem. Duan et al. (2010) used together ant colony optimization and dif-ferential evolution to solve uninhabited combat air vehicle problem. BañOs et al. (2013) proposed a hybrid algo-rithm that combined evolutionary computation and simulated annealing to solve Capacitated Vehicle Routing Problem. Azadeh et al. (2013) presented a particle swarm optimiza-tion algorithm synchronized with a local search heuristic to solve crew scheduling problem. Salcedo-Sanz et al. (2006) suggested two hybrid algorithms by mixing Hopfield neural network with both genetic algorithm and simulated annealing to solve task assignment problem in heterogeneous com-puter systems. Rao and Shyju (2008) presented a hybrid algorithm by mixing simulated annealing and tabu search to solve optimal stacking sequence design problem of laminate composite structures. Gülcü et al.(2016) proposed a hybrid algorithm that combined parallel ant colony optimization and 30 pt. Zeng et al. (2016) suggested an effective hybrid model derived from the combination of differential evolution algorithm with simulated annealing to solve joint replenish-ment and delivery problem. Wang et al. (2015) proposed a hybrid approach that comprised of the adaptive differential evolution algorithm and the back propagation neural net-work to solve time series forecasting problem. Lv et al. (2017) proposed improved fruit fly optimization algorithm based on hybrid location information exchange mechanism

to strengthen basic fruit fly optimization algorithm. Wang et al. (2016) suggested an improved fruit fly optimization algo-rithm that combined the level probability policy method and fruit fly optimization algorithm.

In this paper, there are two main alterations on basic MBO algorithm: multi-flocks structure and interaction between flocks. First change is proposing of multi-flocks structure. Population in basic MBO is created from single flock, whereas population in the IMFMBO algorithm is consist of separately flocks. It is aimed to avoid local optimum with this method. This situation is shown in Fig.3. Second change is, this multi-flocks structure, interaction among each other via flock leaders. This interaction is resemble to interaction among individuals in PSO algorithm. Interaction in proposed algorithm is realized only flocks’ leaders not all individuals. A flock in IMFMBO algorithm resemble to individual in the PSO algorithm and these flocks have a flying speed and position. Leader change is different from the basic MBO, because of interaction of flocks is carried out by the leaders. New leaders’ position is depending on the old leaders’ speed and position. This process is resemble to changing of posi-tion according to speed of individuals in PSO algorithm. New leaders’ position is calculated according to Eq.10. This pro-cess aim to move through to global best region while flocks scanning solutions in search space. Unlike basic MBO, in this method, each bird keep not only solution permutation but also speed and position belonging to solutions. In addition to, while produced neighbor solution, insertion method is used instead of swap method. The swap method can be defined as swapping of two sequence that selected from current solu-tion permutasolu-tion. The insersolu-tion method can be defined as adding a randomly selected sequence from the current

solu-Fig. 4 The solution permutation for 6 cities

tion permutation to a randomly determined position. The solution permutation for six cities is shown in Fig.4. A new permutation obtained by using swap and insertion methods is given Fig.5a, b, respectively. The flowchart of the proposed algorithm is given in Fig.6.

To solve TSP with PSO algorithm, the PSO algorithm must be change to discrete problems. There are some example of solving discrete problems with PSO. Kenedy and Eberhart (1997) developed binary version of PSO for implement the PSO to discrete problems. Wang et al. (2003) and Shi et al. (2007) solved the traveling salesman problems with PSO algorithm. Liao et al. (2007) solved flow shop scheduling problem with PSO algorithm. In their study, solution permu-tation are coded binary. PSO has also been applied to other discrete problems. Tasgetiren et al. (2004b) implemented the discrete PSO algorithm to single machine total weighted tar-diness problem. Tasgetiren et al. (2004a) applied the Smallest Position Value (SPV) rule while determining the solution per-mutation of the solving of Flowshop Sequencing Problem with PSO.

In this paper, MBO and PSO algorithms are used together for solution of TSP. The MBO algorithm is compatible to discrete problems and generally, solution representational is permutation coding shaped. In spite of that, the PSO algo-rithm is compatible for continuous problems and solution of each individual is comprise of continuous values. The TSP is a discrete problem so solution representational of the PSO algorithm should be compatible to discrete prob-lems. According to the PSO algorithm, if ith particle at tth iteration is shown as X_it, solution set of particle Xt_i = [xt

i 1, xi 2t , xi 3t , . . . , xi dt ] for solution of d-dimensional the TSP. Each position in this solution set contains a real value. There-fore, this solution set of particle can not represent a solution permutation. Positions of this particle, which have contin-uous value, must be represent with most frequently used permutation coding or binary coding for implement the PSO algorithm to discrete problems like TSP. In this study, show-ing the PSO solutions are found appropriate permutation coding shaped as well as the MBO algorithm due to the PSO and the MBO algorithms are used together. This alteration is

realized thanks to implementing of the SPV rule as well as Tasgetiren et al. (2004a).

The position (x_{i j f}t ), speed (v_{i j f}t ) and permutation (π_{i j f}t ) information of solution (X_{i f}t ) are given in Table1 for the IMFMBO algorithm, where number of leader change, flock, particle, and dimension are shown t, f, i, and j, respectively. According to SPV rule, position of particle Xt_{i f} is obtain from uniformly distributed values (d uniformly distributed number for d-dimensional problem) between predetermined minimum and maximum values. After that, these values are ranked smallest to biggest. Permutation of particle is formed during this ranking. The x_{i j f}t values are uniformly generated between 0 and 4 for six-dimensional (cities) TSP in Table1. According to the SPV rule, smallest position value (SPV) is x_{i 5 f}t = 0.89, so the dimension j = 5 is assigned to be the first cityπ_{i 1 f}t = 5 in the permutation π_{i f}t . Second smallest position value is x_{i 2 f}t = 1.09, so the dimension j = 2 is assigned to be the second cityπ_{i 2 f}t = 2 in the permutation πt

i f and so on.vi j ft values form uniformly generated values between−4 and 4. The permutation representation of Xt_{i f} particle, which has d = 6, at tth leader change and at f th flock isπ_{i f}t = {5, 2, 3, 4, 1, 6}.

4.1 Initial population of the IMFMBO

PSO algorithm in the IMFMBO algorithm is implemented only among the flock leaders. In other words, PSO algorithm is used for interaction among flocks in proposed algorithm. Therefore, PSO algorithm is implemented to only flock lead-ers. In this way, it is aimed to all flocks move to best global region with forming interaction among flocks. Although PSO is implemented only flock leaders, individuals that except the flock leaders have location, speed, and permutation informa-tion like a PSO particle. Because each bird is a potential leader.

Population is same in each flock, and this is equal to p. A bird in any flock is also a particle. After that, particle or bird defined as individual. Initial position and speed values which determine to initial solution permutation of each individual in each flock are determined to Eqs.7and8, respectively. x_{i j f}0 = xmin+ (xmax− xmin) ∗ r1 (7)

i = 1, 2, …, p; j = 1, 2, …, d; f = 1, 2, …, z.

PSO-based improved multi-flocks migrating birds optimization (IMFMBO) algorithm for solution... 5475

Fig. 6 IMFMBO flowchart

where p is define the number of individuals in a flock, d is define the problem dimensions (number of cities in TSP) and z is define the number of flocks. x_{i j f}0 is defined as position of ith individual in the first leader in f th flock at jth dimen-sion. Also xmin= 0, xmax= 4 and r1is randomly uniform

distributed value at [0,1].

v0

i j f = vmin+ (vmax− vmin) ∗ r2 (8)

wherev_{i j f}0 defined as speed of ith individual in the first leader, f th flock at jth dimension.vmin = − 4, vmax = 4 and r2is

randomly uniform distributed value at [0,1].

Initial permutations of individuals that have the position information, obtain according to SPV rule.

Table 1 Solution presentation of particle Xt i f Dimension, j 1 2 3 4 5 6 x_{i j f}t 2.63 1.09 2.20 2.53 0.89 2.93 vt i j f −0.13 0.19 2.23 −3.5 −3.81 1.30 πt i j f 5 2 3 4 1 6

Table 2 Neighborhood applied to permutation before repairing

Dimension, j 1 2 3 4 5 6 xt i j f 2.63 1.09 2.20 2.53 0.89 2.93 πt i j f 5 2 3 4 1 6 xt i j f 2.63 1.09 2.20 2.53 0.89 2.93 πt i j f 2 3 4 5 1 6

Number of pBest in the IMFMBO algorithm is different

from basic PSO and this value is equal to flock amount. Because PSO algorithm is applied only flock leaders in pro-posed algorithm. Therefore, each flock have a pBest value

and initial value is determined as leaders’ position for each flock. In other words, p_Bestf = X0_{1 f}. The global best (gBest)

is determined as selected best pBestfrom all pBestvalues.

Flock leader in MBO algorithm is quietly active because of neighbor sharing to both side of V-shape. Therefore, the interaction of the flocks is carried out only through flock leaders and according to Eqs.9and10, these leaders must have a velocity vector. Initial value of this velocity vector is equal to velocity vector of flock leader for each flock and determined to Eq.8. In other words,∂0_f = v_{1 f}0 , f = 1, 2,…, z, where∂0_f is the initial velocity vector of f th flock. Then, this velocity vector is updated according to Eq.9in each leader change.

4.2 Neighborhood of the IMFMBO

Neighborhood generation in the IMFMBO algorithm, is applied to permutations of individuals directly. Insertion method is chosen as neighborhood method. However, there is need a repair process due to permutation of individuals are constituted according to SPV rule. This approach is shown in Tables2 and3, where cityπ_{i 1 f}t = 5 is inserted to city πt

i 4 f = 4.

As shown in Table2, if neighborhood generation is apply to permutations directly, SPV rule will broke. Position of generated new individual should be corrected in order to generated neighbor solution be suitable to SPV rule. In Table 2, position x_{i 1 f}t = 2.63 at j = 1 is inserted to j = 4 due to repair process. Thus, SPV rule is pro-tected.

Table 3 Neighborhood applied to permutation after repairing

Dimension, j 1 2 3 4 5 6 xt_{i j f} 2.63 1.09 2.20 2.53 0.89 2.93 πt i j f 5 2 3 4 1 6 xt_{i j f} 1.09 2.20 2.53 2.63 0.89 2.93 πt i j f 2 3 4 5 1 6

4.3 Interaction of flocks by using PSO

In the IMFMBO algorithm, a multi-flock structure is used to strengthen the global search. These flocks are created independently of each other at the beginning. In order to further strengthen the global search, a method of interac-tion between the flocks is used. The PSO algorithm is used for this interaction. The PSO algorithm is inspired by the observations of birds and fish. These flocks are often exhib-ited behavior synchronize influenced by each other in search of food. These synchronized behaviors are mathematically expressed as in Eqs.5 and6 in the PSO algorithm. Every individual in the PSO algorithm has a velocity and posi-tion informaposi-tion. Via this informaposi-tion, the next posiposi-tion of each individual is calculated. In this calculation, besides the previous velocity and position information of the indi-viduals, the best solutions they have found so far are also playing an important role. In the PSO algorithm, the best solution found in the flock is considered to be the global solu-tion and all the individuals in the flock are directed toward this global solution. The basic MBO algorithm is also an algorithm based on the logic of the flock. But, its working principle is different from PSO. In the basic MBO algo-rithm, the interaction between the individuals in the flock is provided by the sharing mechanism. The neighbors of the leader solution are shared on both sides of the V-shaped flock. Therefore, the leader solution much more influence the direction of motion in search space than other solutions. Therefore, the PSO method, which is used for the interac-tion of flocks in the IMFMBO algorithm, is applied only to flock leaders. The interaction of the flocks is given in Fig.7.

As shown in Fig.7, every leader in the flock can be thought of as an individual in the PSO. The flock leader’s new posi-tion is determined using the velocity, posiposi-tion and current best solution information. The purpose of this is to bring the flocks closer to the global solution, as in the PSO algorithm. The interaction between the flock leaders is only implemented during the leader change process. Thus, the structure of the basic MBO algorithm is preserved and every flock continues the local search in its own area until the leader change takes place.

PSO-based improved multi-flocks migrating birds optimization (IMFMBO) algorithm for solution... 5477

Fig. 7 Interaction of flocks by using PSO

Interaction of flocks is realized according to Eqs.9and

10. ∂t+1 f = w∂ t f + c1r1(pBestf − X t 0 f) + c2r2(gBest− Xt0 f) (9) Xt_{0 f}+1= Xt_{0 f} + ∂t_f+1 (10) where ∂t_f+1 is represent the velocity vector of f th flock in t+1th leader change, w is inertia factor, ∂t_f is the velocity vector of f th flock in the tth leader change, c1

and c2 are learning factors and generally their values

are equal to 2. r1 and r2 are uniformly distributed

ran-dom variables in [0, 1] range. p_Bestf is the position of best solution that found by f th flock leader till now, Xt_{0 f} is the position of previous leader in f th flock, and gBest is the position of best solution that found by all

flocks.

According to Eq.9, velocity vector of flock f is updated for position of new leader (Xt_{0 f}+1). According to Eq. 10, position of new leader is updated with new velocity vector and previous leader position. A new permutation is obtained depending on the new position and so a new solution is obtained.

Algorithm of the IMFMBO is given in Algorithm 2 for minimization problems.

According to SPV rule, generate z flocks with p initial solutions randomly for each flock and place them on V formation ({S11, S21, . . . , Sp1}, {S12, S22, . . . , Sp2}, . . ., {S1z, S2z, . . . ,

Spz})

set velocities, pBest, gBestfor each flock leader repeat

Apply MBO algorithm for each flock until leader replacing Replace leader for each flock

for f = 1 to z do

according to Equation9and10, update∂t_f+1and Xt_{0 f}+1, respectively

if f(Xt_{0 f}+1) < f (pBest) thenf

p_Bestf = Xt_{0 f}+1

end if end for

Find(pBest)mi nfrom all pBest

if ( f((pBest)mi n) < f (gBest)) then

g_Best= (pBest)mi n

end if until stop criteria

Get best result from all flocks

Algorithm 2: IMFMBO algorithm

5 Experimental results

In this paper, the basic MBO and the IMFMBO algorithms are tested with 22 instances that chosen from TSPLIB (Reinelt

Table 4 Initial, increment and end values for parameter settings Parameter Initial value Increment End value

Bird (p) 11 20 91 Neighbor (k) 3 2 21 Flap (m) 10 10 50 Shared neighbor (x) 1 1 (k− 1)/2 Flock (z) 1 1 15 c1 1 0.5 2 c2 1 0.5 2

Table 5 Obtained best parameter values for TSP

Parameter Best value

Bird (p) 71 Neighbor (k) 3 Flap (m) 30 Shared neighbor (x) 1 Flock (z) 15 c1 1 c2 1

1991). Lots of instances in the TSPLIB are already solved in the literature, and their best know solutions are used for com-pared the algorithms. These instances have cities between 51 and 654. All of them are member of the Euclidean distance type in Reinelt (1991). A TSP instance includes cities and coordinate of cities in its content. Numerical value in the instance name is represent the number of cities, for example, if an instance named as berlin52, this means, instance has 52 cities.

Both algorithms, the basic MBO, and the IMFMBO, have been performed on a Intel(R) Core(TM) i5-3330 CPU @ 3.00GHz processor, 4 GB RAM and Linux Ubuntu 14.04 (64-bit) operating system.

5.1 Parameter settings

In this section, appropriate parameters of the proposed algorithm and the MBO algorithm are tried to found for sym-metric TSP. Initial value, end value and increment amounts of each parameter are given in Table4. In Table4, the bird parameter is investigated in five combinations with 20 incre-ment values between 11 and 91. The neighborhood parameter is investigated in 10 combinations with two increment val-ues between 3 and 21. Similarly, flap and flock parameters are examined with five combinations and 15 combinations, respectively. Unlike the other parameters, the end value of the sharing parameter depends on the neighborhood param-eter [(k − 1)/2]. Therefore, the number of combinations varies according to the state of k. Unlike the basic MBO, the

Table 6 Computational results of the IMFMBO algorithm for 22 TSP instances from TSPLIB

Instance Cities Opt Best (%) Average (%) Worst (%)

Eil51 51 426 0 0.3912 1.1737 Berlin52 52 7542 0 0.3504 3.4075 St70 70 675 0 1.0913 2.9629 Eil76 76 538 0 1.5241 3.5315 Pr76 76 108159 0 1.2365 2.9891 KroA100 100 21282 0 2.6769 6.1601 KroB100 100 22141 0.3568 2.9438 5.3746 Eil101 101 629 0.7949 2.9093 3.9745 Lin105 105 14379 2.8096 5.2043 7.5526 Pr124 124 59030 0.8283 2.0491 3.5863 Bier127 127 118282 1.0035 4.2200 6.1184 Ch130 130 6110 0.8346 3.9470 6.1702 Pr136 136 96772 3.6188 6.3571 8.8672 Ch150 150 6528 1.5625 4.2248 6.0661 KroA150 150 26524 2.6089 5.0196 7.3593 KroB150 150 26130 2.4263 4.4228 6.3834 U159 159 42080 0.4800 4.7647 9.0850 KroA200 200 29368 4.9067 7.0349 8.6999 Tsp225 225 3919 4.1592 6.5969 8.5225 A280 280 2579 8.1039 11.4462 14.8894 Fl417 417 11861 12.1996 17.1404 22.6793 P654 654 34643 21.3260 25.8368 28.4155

IMFMBO’s c1and c2 parameters are investigated in three

combinations with 0.5 increment values between 1 and 2. In this case, the number of all combinations is 20,760 (5× 10 × [(k − 1)/2] × 5 × 15 × 3 × 3 = 20,760). In addition to, the proposed method is run five times with same parameters at each combination and average of results are calculated. Therefore, the proposed method is run 103,800 (20,760× 5) times in total. Obtained results are evaluated and best param-eters for TSP are given in Table5. Both algorithm are run with p = 71, k = 3, m = 30, and x = 1 parameter values in Table 5. In addition to these parameters, the IMFMBO algorithm is run with z= 15, c1= 1 and c2= 1 parameters.

Algorithms are run independently 30 times with these param-eter values in each case study. Number of iteration of both algorithm are determined as; number of generated neighbor solution = (dimension of problem)3_.

5.2 Experiments

Summary of experimental results is given in Table6, where first column shows the name of instance. Column ’Cities’ shows the number of cities. Column ’Opt’ shows the distance of optimal results in TSPLIB. Column ’Best’ shows error of the distance of best solution that found by the IMFMBO

PSO-based improved multi-flocks migrating birds optimization (IMFMBO) algorithm for solution... 5479

Table 7 Comparison of both algorithms, the basic MBO and the IMFMBO on 22 TSP instances from TSPLIB

Instance Basic MBO IMFMBO

Best Average Worst Best Average Worst

Eil51 441 465.4000 491 426 427.6666 431 Berlin52 8114 8523.1670 9193 7542 7568.4333 7799 St70 743 823.4000 922 675 682.3666 695 Eil76 583 620.5667 654 538 546.2000 557 Pr76 122862 130258.7000 140751 108159 109496.467 111392 KroA100 24591 29716.1300 33343 21282 21851.7000 22593 KroB100 25703 30288.1700 36143 22220 22792.8000 23331 Eil101 703 744.6333 794 634 647.3000 654 Lin105 18798 21071.8700 24809 14783 15127.3333 15465 Pr124 81623 98647.3000 114464 59519 60239.63333 61147 Bier127 141340 152450.8000 168914 119469 123273.5670 125519 Ch130 7905 8522.3000 9197 6161 6351.1666 6487 Pr136 124389 142910.3000 155380 100274 102923.9330 105353 Ch150 8499 9460.6000 10314 6630 6803.8000 6924 KroA150 33522 40307.9700 47236 27216 27855.4000 28476 KroB150 34297 40706.6300 46068 26764 27285.7000 27798 U159 58022 68255.2300 76714 42282 44085.0000 45903 KroA200 42636 48423.1300 54647 30809 31434.0333 31923 Tsp225 5702 6157.5670 6810 4082 4177.5333 4253 A280 4197 4607.0330 5243 2788 2874.2000 2963 Fl417 47487 57292.1 75325 13308 13894.0333 14551 P654 181616 225682.8300 274856 42031 43593.6667 44487

algorithm, column ’Worst’ shows error of the distance of the worst solution that found by the IMFMBO algorithm and column ’Average’ shows average error of the distance of solution that found by the IMFMBO algorithm. Error is the percentage value and is calculated as in Eq.11.

error= result− opt

opt × 100 (11)

Comparison of experimental results that obtained from the basic MBO and the IMFMBO algorithm is given in Table7. As is seen from this table, obtained better solution all of 22 instances in the IMFMBO algorithm compared to the basic MBO. As shown in Tables6 and 7, known best solutions are obtained in six instance by the IMFMBO algorithm. And close to solutions to best solutions are obtained in rest of solutions.

Summary of experimental convergence graphs of the basic MBO and the IMFMBO for 22 TPS instances is shown in Figs. 8 and 9. Both algorithms are run 30 times independently for 22 TSP instances, and average of minimum results are calculated at each leader change. As shown in Figs. 8 and 9, convergence rate of the IMFMBO algorithm is same with convergence rate of the MBO algorithm. Nevertheless, better results are obtained

in the IMFMBO algorithm compared to the MBO algo-rithm.

In this paper, the performance of the IMFMBO algo-rithm is compared with algoalgo-rithms in the literature such as Discrete Artificial Bee Colony Algorithm (DABC) (Kıran et al. 2013), Improved Ant Colony Optimization (IACO) (Tuba and Jovanovic 2013), Discrete Invasive Weed Opti-mization (DIWO) (Zhou et al. 2015), Particle Swarm Optimization (PSO) (Shi et al. 2007), Hybrid Differen-tial Evolution Algorithm (HDE) (Wang and Xu 2011) and Improved Fruit Fly Optimization Algorithm (IFOA) (Huang et al. 2017). The values in Tables 8 and9 show the error (according to Eq. 11) of the distance of best solution obtained by these algorithms. ’NA’ indicates that there is not the result of the problem in the relevant refer-ence.

Tables8and9contain 11 and 10 instances, respectively. The proposed algorithm has found the optimal solution in 6 problems both of them. In the instances of Tables8and9, for 2 problems, proposed algorithm has found error less than 1%. As shown in Tables 8and9, the IMFMBO algorithm has achieved the optimal result in small-sized problems. The IMFMBO algorithm has produced very close results to opti-mal value despite the increase in the problem dimension.

PSO-based improved multi-flocks migrating birds optimization (IMFMBO) algorithm for solution... 5481

Table 8 Comparison of the IMFMBO algorithm with other algorithms selected from the literature (DIWO, IACO, DABC)

Instance Opt DIWO (Zhou et al.2015) IACO (Tuba and Jovanovic2013) DABC (Kıran et al.2013) IMFMBO

Best Error (%) Best Error (%) Best Error (%) Best Error (%)

Eil51 426 428 0.4694 427 0.2347 428 0.4694 426 0 Berlin52 7542 7544 0.0265 7542 0 7544 0.0265 7542 0 St70 675 677 0.2962 675 0 677 0.2962 675 0 Eil76 538 NA NA 538 0 NA NA 538 0 Pr76 108159 NA NA 108358 0.1839 108159 0 108159 0 KroA100 21282 NA NA 21282 0 21285 0.0140 21282 0 Eil101 629 NA NA NA NA 653 3.8155 634 0.7949 Lin105 14379 NA NA 14379 0 NA NA 14783 2.8096 Pr124 59030 NA NA 59030 0 NA NA 59519 0.8283 Pr136 96772 NA NA 96781 0.0093 NA NA 100274 3.6188 A280 2579 NA NA NA NA 2818 9.2671 2788 8.1039

Table 9 Comparison of the IMFMBO algorithm with other algorithms selected from the literature (PSO, HDE, IFOA)

Instance Opt PSO (Shi et al.2007) HDE (Wang and Xu2011) IFOA (Huang et al.2017) IMFMBO

Best Error (%) Best Error (%) Best Error (%) Best Error (%)

Eil51 426 427 0.2347 439 3.0516 426 0 426 0 Berlin52 7542 7542 0 7542 0 7542 0 7542 0 St70 675 675 0 684 1.3333 675 0 675 0 Eil76 538 546 1.4869 558 3.7174 540 0.3717 538 0 Pr76 108159 108280 0.1118 109491 1.2315 NA NA 108159 0 KroA100 21282 NA NA NA NA 21282 0 21282 0 KroB100 22141 NA NA NA NA 22219 0.3522 22220 0.3568 Eil101 629 NA NA NA NA 653 0.9538 634 0.7949 Lin105 14379 NA NA NA NA 14379 0 14783 2.8096 Ch150 6528 NA NA NA NA 6558 0.4595 6630 1.5625

Boldface letter denotes the best results obtained by the algo-rithms.

6 Discussion and conclusion

The MBO algorithm is a metaheuristic search algorithm that is inspired by V formation during the migration of migra-tory birds. The MBO aim to develop current solutions in every step after created of initial population. Neighborhood structure is used like lots of metaheuristic algorithm for local search. The sharing mechanism helps to improve weak solu-tions. But, the MBO algorithm can be easily down to local optima. In this paper, a new approach has been proposed to overcome disadvantages of the MBO.

The basic MBO algorithm is improved by introducing two main innovations to solve discrete problems like TSP. First, the multi-flock structure instead of a single flock is used to avoid to the local optimum. Secondly, the PSO

algorithm at interaction among these multi-flocks is used to reinforce the global search. Also, structure of insertion neigh-borhood is used in the IMFMBO instead of swap. Because this alteration is more effective in TSP. The IMFMBO is implemented on twenty two TSP instances. The performance of the IMFMBO algorithm is compared with the basic MBO and some selected metaheuristic algorithms in the literature. The obtained results showed that the IMFMBO algorithm outperforms the basic MBO algorithm in all cases.

Experimental studies show that the IMFMBO algorithm often achieves the best solution. In addition, close results are obtained by the IMFMBO algorithm when looking at the average values for each problem. Average and best val-ues show that as the number of cities increases, is moved away from the optimal solutions. On the other hand, sat-isfactory results are obtained on some complex examples. When the IMFMBO and the basic MBO are compared, it is seen that the IMFMBO generates the better solutions in all of the instances. Even the worst results of the IMFMBO are

PSO-based improved multi-flocks migrating birds optimization (IMFMBO) algorithm for solution... 5483

better than the best results of the basic MBO. Compared to the IMFMBO and other algorithms in the literature, it has been generated better results than DIWO in all cases. When the proposed algorithm is compared with IACO, it is seen that the results obtained from both algorithms are close to each other. In addition to this, the problem dimension appears to have affected the result. When the proposed algorithm is compared with DABC, PSO and HDE, it is seen that the IMFMBO equal or better results for all problems. When the IMFMBO algorithm is compared with the IFOA algorithm, it is seen that the proposed algorithm finds more solutions than the IFOA according to the number of obtained optimal solutions in the cases where the optimal solutions are found. In some problems where optimal values cannot be obtained, the IFOA algorithm has produced slightly more near values than the proposed algorithm. In addition, there are problems that the proposed algorithm outperforms IFOA. According to all experimental results, the proposed algorithm is generally sensitive to the problem dimension. As the problem dimen-sion increases, the IMFMBO is finding it difficult to find the optimal solution. The IMFMBO produced better results than the basic MBO in all cases. These results have proven that the modifications to the basic MBO are successful.

In the future works, the performance of the IMFMBO algorithm on flow shop scheduling and cutting problems will be investigated. In addition, the performance of the MBO and the IMFMBO algorithms on continuous functions can be examined.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict of interest.

References

Azadeh A, Farahani MH, Eivazy H, Nazari-Shirkouhi S, Asadipour G (2013) A hybrid meta-heuristic algorithm for optimization of crew scheduling. Appl Soft Comput 13(1):158–164

BañOs R, Ortega J, Gil C, MáRquez AL, De Toro F (2013) A hybrid meta-heuristic for multi-objective vehicle routing problems with time windows. Comput Ind Eng 65(2):286–296

Debels D, De Reyck B, Leus R, Vanhoucke M (2006) A hybrid scat-ter search/electromagnetism meta-heuristic for project scheduling. Eur J Oper Res 169(2):638–653

Dorigo M, Birattari M, Stutzle T (2006) Ant colony optimization. IEEE Comput Intell Mag 1(4):28–39

Duan H, Yu Y, Zhang X, Shao S (2010) Three-dimension path plan-ning for ucav using hybrid meta-heuristic aco-de algorithm. Simul Model Pract Theory 18(8):1104–1115

Duman E, Elikucuk I (2013) Solving credit card fraud detection problem by the new metaheuristics migrating birds optimization. In: Inter-national work-conference on artificial neural networks. Springer, pp 62–71

Duman E, Uysal M, Alkaya AF (2012) Migrating birds optimization: a new metaheuristic approach and its performance on quadratic assignment problem. Inf Sci 217:65–77

Dutta M et al (2015) TSP solution using dimensional ant colony opti-mization. In: 2015 Fifth international conference on advanced computing & communication technologies (ACCT). IEEE, pp 506–512

Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micro machine and human science, 1995. MHS’95. IEEE, pp 39– 43

Gao KZ, Suganthan PN, Chua TJ (2013) An enhanced migrating birds optimization algorithm for no-wait flow shop scheduling prob-lem. In: 2013 ieee symposium on computational intelligence in scheduling (SCIS). IEEE, pp 9–13

Geng X, Chen Z, Yang W, Shi D, Zhao K (2011) Solving the traveling salesman problem based on an adaptive simulated annealing algo-rithm with greedy search. Appl Soft Comput 11(4):3680–3689 Gülcü ¸S, Mahi M, Baykan Ö.K, Kodaz H (2016) A parallel

cooper-ative hybrid method based on ant colony optimization and 3-opt algorithm for solving traveling salesman problem. Soft Comput 1–17

Helsgaun K (2000) An effective implementation of the Lin–Kernighan traveling salesman heuristic. Eur J Oper Res 126(1):106–130 Kindly provide volume number for the references Gulcu et al. (2016) and Lv et al. (2017)

Huang L, Wang Gc, Bai T, Wang Z (2017) An improved fruit fly opti-mization algorithm for solving traveling salesman problem. Front Inf Technol Electron Eng 18(10):1525–1533

Jolai F, Rabiee M, Asefi H (2012) A novel hybrid meta-heuristic algo-rithm for a no-wait flexible flow shop scheduling problem with sequence dependent setup times. Int J Prod Res 50(24):7447–7466 Karapetyan D, Gutin G (2011) Lin–Kernighan heuristic adaptations for the generalized traveling salesman problem. Eur J Oper Res 208(3):221–232

Kennedy J, Eberhart RC (1997) A discrete binary version of the parti-cle swarm algorithm. In: 1997 IEEE International Conference on Systems, Man, and Cybernetics, Computational Cybernetics and Simulation, 1997. IEEE, vol. 5, pp 4104–4108

Kıran MS, ˙I¸scan H, Gündüz M (2013) The analysis of discrete artifi-cial bee colony algorithm with neighborhood operator on traveling salesman problem. Neural Comput Appl 23(1):9–21

Liao CJ, Tseng CT, Luarn P (2007) A discrete version of particle swarm optimization for flowshop scheduling problems. Comput Oper Res 34(10):3099–3111

Liu F, Zeng G (2009) Study of genetic algorithm with reinforcement learning to solve the TSP. Expert Syst Appl 36(3):6995–7001 Lv SX, Zeng YR, Wang L (2017) An effective fruit fly optimization

algorithm with hybrid information exchange and its applications. Int J Mach Learn Cybern 1–26

Ma J, Yang T, Hou ZG, Tan M, Liu D (2008) Neurodynamic pro-gramming: a case study of the traveling salesman problem. Neural Comput Appl 17(4):347–355

Marinakis Y, Marinaki M, Migdalas A (2016) A hybrid discrete artificial bee colony algorithm for the multicast routing problem. In: Euro-pean conference on the applications of evolutionary computation. Springer, pp 203–218

Meer K (2007) Simulated annealing versus metropolis for a TSP instance. Inf Process Lett 104(6):216–219

Niroomand S, Hadi-Vencheh A, ¸Sahin R, Vizvári B (2015) Modified migrating birds optimization algorithm for closed loop layout with exact distances in flexible manufacturing systems. Expert Syst Appl 42(19):6586–6597

Pan QK, Dong Y (2014) An improved migrating birds optimisation for a hybrid flowshop scheduling with total flowtime minimisation. Inf Sci 277:643–655

Poli R, Kennedy J, Blackwell T (2007) Particle swarm optimization. Swarm Intell 1(1):33–57

Poorzahedy H, Rouhani OM (2007) Hybrid meta-heuristic algorithms for solving network design problem. Eur J Oper Res 182(2):578– 596

Rao ARM, Shyju P (2008) Development of a hybrid meta-heuristic algorithm for combinatorial optimisation and its application for optimal design of laminated composite cylindrical skirt. Comput Struct 86(7):796–815

Reinelt G (1991) Tspliba traveling salesman problem library. ORSA J Comput 3(4):376–384

Salcedo-Sanz S, Xu Y, Yao X (2006) Hybrid meta-heuristics algorithms for task assignment in heterogeneous computing systems. Comput Oper Res 33(3):820–835

Shi XH, Liang YC, Lee HP, Lu C, Wang Q (2007) Particle swarm optimization-based algorithms for TSP and generalized TSP. Inf Process Lett 103(5):169–176

Tasgetiren MF, Sevkli M, Liang YC, Gencyilmaz G (2004a) Par-ticle swarm optimization algorithm for permutation flowshop sequencing problem. In: International workshop on ant colony optimization and swarm intelligence. Springer, pp 382–389 Tasgetiren MF, Sevkli M, Liang YC, Gencyilmaz G (2004b) Particle

swarm optimization algorithm for single machine total weighted tardiness problem. In: Congress on evolutionary computation, 2004, CEC2004. IEEE, vol. 2, pp 1412–1419

Tongur V, Ülker E (2014) Migrating birds optimization for flow shop sequencing problem. J Comput Commun 2(04):142

Tongur V, Ülker E (2016) The analysis of migrating birds optimiza-tion algorithm with neighborhood operator on traveling salesman problem. In: Intelligent and evolutionary systems. Springer, pp 227–237

Tuba M, Jovanovic R (2013) Improved ACO algorithm with pheromone correction strategy for the traveling salesman problem. Int J Com-put Commun Control 8(3):477–485

Wang X, Xu G (2011) Hybrid differential evolution algorithm for trav-eling salesman problem. Procedia Eng 15:2716–2720

Wang KP, Huang L, Zhou CG, Pang W (2003) Particle swarm opti-mization for traveling salesman problem. In: 2003 International conference on machine learning and cybernetics. IEEE, vol. 3, pp 1583–1585

Wang Z, Geng X, Shao Z (2009) An effective simulated annealing algorithm for solving the traveling salesman problem. J Comput Theor Nanosci 6(7):1680–1686

Wang L, Zeng Y, Chen T (2015) Back propagation neural network with adaptive differential evolution algorithm for time series forecast-ing. Expert Syst Appl 42(2):855–863

Wang L, Liu R, Liu S (2016) An effective and efficient fruit fly optimiza-tion algorithm with level probability policy and its applicaoptimiza-tions. Knowl-Based Syst 97:158–174

Zeng YR, Peng L, Zhang J, Wang L (2016) An effective hybrid differ-ential evolution algorithm incorporating simulated annealing for joint replenishment and delivery problem with trade credit. Int J Comput Intell Syst 9(6):1001–1015

Zhong WH, Zhang J, Chen WN (2007) A novel discrete particle swarm optimization to solve traveling salesman problem. In: IEEE congress on evolutionary computation, 2007, CEC 2007. IEEE, pp 3283–3287

Zhong Y, Wu C, Li L, Ning Z (2008) The study of neighborhood structure of tabu search algorithm for traveling salesman prob-lem. In: Fourth international conference on natural computation, 2008, ICNC’08. IEEE, vol. 1, pp 491–495

Zhou Y, Luo Q, Chen H, He A, Wu J (2015) A discrete invasive weed optimization algorithm for solving traveling salesman problem. Neurocomputing 151:1227–1236

Publisher’s Note Springer Nature remains neutral with regard to juris-dictional claims in published maps and institutional affiliations.