TÀI LIỆU THAM KHẢO-Solving Travelling Salesman Problem Using variants of ABC Algorithm

01, M arch 2013 ISSN – 2278-1080 Th e In tern atio n al Jo urn al o f C o m puter Scien ce & Application s TIJCSA RESEARCH PAPER Available O n lin e at h ttp://w w w .journ alofcom put

Vo lum e 2, N o 01, M arch 2013 ISSN – 2278-1080

Th e In tern atio n al Jo urn al o f C o m puter Scien ce &

Application s (TIJCSA) RESEARCH PAPER Available O n lin e at

h ttp://w w w journ alofcom puterscien ce.com /

Solving Travelling Salesman Problem Using

Variants of ABC Algorithm

Ginnu George

Student, Computer Science and Engineering

Karunya University ginnu89@gmail.com

Dr.Kumudha Raimond

Professor, Computer Science and Engineering

Karunya University kraimond@karunya.edu

Abstract

This paper mainly explains about the performance of variants of Artificial Bee Colony (ABC) algorithms in solving the Travelling Salesman Problem (TSP) The main goal of TSP is that a number of cities should be visited by a salesman and return to the starting city along with a number of possible shortest routes In this work, it investigates over variants of ABC algorithms such as Improved ABC (I-ABC) and Prediction Selection ABC (PS-ABC) The variants of ABC algorithms are used to find the optimal path for TSP The results of the original ABC algorithms are compared with the results of the I-ABC and PS-ABC algorithms and shows that the PS-ABC performs well in finding the shortest distance within the minimum span of time

Keywords- Artificial Bee Colony, Improved ABC, Prediction Selection ABC

1 Introduction

Optimization problems is mainly used for finding nearly optimal solution and are frequently encountered in various applications such as TSP [10], Container Loading problems (CL) [11], Scheduling problems [12], engineering design [13, 14] etc TSP is one of the NP hard optimization problems In TSP, the salesman travels all the cities at once and returns to the starting city with the possible shortest route within small duration Many heuristic optimization methods are developed so far for searching nearly optimal solution in solving TSP such as Genetic Algorithm (GA) [1, 2], Particle Swarm Optimization (PSO) [3, 5], Ant Colony optimization (ACO) [4, 6], Simulated Annealing (SA) [7] and Artificial Bee Colony (ABC) [8,

9] ABC algorithm shows more effective when compared to the above optimization algorithms since it is the latest optimization technique which is developed by Dervis Karaboga in the year

2005 In the year 2012 the variants of ABC Algorithms [ 17] such as I-ABC and PS-ABC has been proposed and shows that PS-ABC performs well in finding the nearly optimal solution within minimum span of time This work investigates the variants of ABC algorithms over TSP inorder to find the trend of PS-ABC algorithm with respect to original ABC and I-ABC algorithm

The outline of the paper is organized as follows: section 2 explains the concepts of TSP, section

3 explains the architecture of variants of ABC algorithms, the results and discussions in solving TSP are presented in section 4 and section 5 concludes the paper

2 Travelling Salesman Problem (TSP)

The main goal of TSP is that, the salesman travels through a number of cities, visits each city once and returns to the starting city as trying to obtain a closed tour in finding a shortest path within minimum span of time The distance between the two cities i and i+1 is calculated with help of the following eq (1) [15]:

)= + ] 1 [ ], [ (T i T i

The total tour length [15] is calculated in the eq (2):

=

f ∑−

=

+ +

1

1 ( [ ], [ 1 ] ( [ ], [ 1 ])

n

Where, ‘n’ is the total number of cities

‘T’ is the tour length

In solving the TSP Nearest Neighbor method [16] is used The Nearest Neighbor method compares the distribution of distances in which from one point to nearest neighbor point in the given set of locations The main steps of Nearest Neighbor Method are as follows:

Step 1: Initially starts from a random node

Step 2: Moves towards the unvisited node

Step 3: Repeat the above steps until all nodes are visited and finally joins the first and last node

to form a closed route and hence obtains the reference path

3 Artificial Bee Colony Algorithm and its Variants ( I-ABC and PS-ABC )

The I-ABC and PS-ABC are the variants of ABC algorithms The following subsections explain

in detail about the Original ABC algorithm and its variants

3.1 Original ABC Algorithm

In ABC algorithm, the food source position represents a solution to the optimization problem and the nectar amount of a food source corresponds to the fitness of the corresponding solution The following Fig 1 [18] explains about the original ABC algorithms

STEP 1: At the first step, the ABC generates a randomly distributed initial population of solutions (SN), where SN denotes the size of employed bees or onlooker bees Each solution Xi

is a D-dimensional vector where i=1,2,…SN and D is the number of optimization parameters STEP 2: After initialization, the initial fitness of the population is evaluated The population of the solutions is then subjected to repeated cycles such as employed bees, the onlooker bees and the scout bees

STEP 3: For each employed bee new solutions (Vij) is produced by using the solution search equation

∑ =

= SN

j fit j

fiti pi

Where, fiti denotes the fitness value of the ith solution

Solution search equation is given below (4) [17]:

) (xij xkj

ij xij

where k is {1,2,… ,SN} and j is {1,2……,D} are randomly generating indexes, Φij is a random

number between [−1, 1] and Xij is the food position or the solution After obtaining the new solution its fitness is evaluated and then it applies the greedy solution mechanism that is if the fitness value of the new one is better than that of the previous one, then the employed bee would memorize the new position and forget the previous one Otherwise it keeps the position of the

previous one in its memory For each employed bee new solutions (Vij) is produced by using the solution search equation shown eq (4).After obtaining the new solution its fitness is evaluated and finally applies the greedy solution mechanism

STEP 5: If a position cannot be improved further through a predetermined number of cycles, the food source should be abandoned Determine the abandoned solution for the scout, if exists, and replace it with a new randomly produced solution xij

STEP 6: Memorize the best solution that is obtained so far

STEP 7: Repeat the cycle until the termination condition is satisfied The main strength of ABC

is having good exploration while the limitations are slow convergence and poor exploitation

Fig.1 Flowchart of original ABC algorithm [18]

3.2 Improved ABC (I-ABC)

To overcome the limitations of original ABC, Improved ABC (I-ABC) is proposed In I-ABC inertia weight and acceleration coefficients are introduced in the solution search equation to modify the search process

v ij=x ij w ij+ 2 (φij− 0 5 )(x ij−x kj)Φ1 +ϕij(x j−x kj)Φ2 (5)

Where, ωij is the inertia weight, xj is the jth parameter of the best-so-far solution, Φij and φij are

random numbers between [0, 1], Φ1 2

maximum step size The inertia weight and acceleration coefficients are defined as functions of the fitness in the search process of ABC They are given in the following equations (6), (7) and (8) [17]:

ap i fitness

wij

− +

=

2 1 exp( fitness )1 (i ap

− +

=

Φ if bee is looked one (8)

Where, ap is the fitness value in the first iteration

3.3 Prediction Selection ABC (PS-ABC)

To increase the exploitation capacity and convergence speed and also to overcome the trapping

of local optimal solutions in Improved ABC (I-ABC), a high efficient algorithm, called PS-ABC (Prediction-Selection ABC) algorithm In PS-ABC [17], an employed bee firstly works out three new solutions by three different solution search equations, and then chooses and finally determines the best one as the candidate solution The Solution Search Equations used in

PS-ABC are: the first one is eq (4), which is the solution modification form of the original PS-ABC

algorithm The second one is the eq (5) and the third one is the GABC equation which is

explained by the following eq (9)

vij = xij+ 2 (φij− 0 5 )(xij−xkj) +ϕij(x j−xkj) (9)

Where, vij is the new feasible solution that is an modified feasible solution depending on its previous solution xij xj is the jth parameter of the best-so-far solution, Φij is a random number between [0, 1], φij is between [0, c], c is a non negative constant, which is set 1.The Fig 2 shown below explains the explains the main operations of PS-ABC

Fig 2 Flowchart of PS-ABC [17]

3.4 Architecture of ABC Algorithm for TSP

Fig.3 shows the flowchart of ABC algorithm for optimization of the TSP [16] In the initialization phase, the control parameters are set, such as colony size, iteration number etc In the next phase, a reference path is obtained by using nearest neighbor method When the working bees are initialized, the bee optimization loop is set Then the random node is assigned for the bee and then computes the probabilities by the eq (3) and calculates the path and will memorize the best solution by applying greedy selection mechanism Finally the bees will become as scout

bees and the working bees are updated The optimization loop is terminated when the

termination criterion is satisfied and hence the best solution is obtained

No Yes

Fig.3 Flowchart of ABC algorithm for Travelling Salesman problem [16]

Stop

Start

Set control parameters

Obtain no of locations

Compute the path using nearest neighbor method

Initialize the working bees

Set Cycle =1 Reset the path

Construct the new paths for

bees Compute the step probabilities Compute fitness

Memorize best solution

Cycle=cycle+1

4 Results and Discussions

In TSP the colony size is taken as 40 The number of cities is taken as 50 and the number of iteration is 50 Fig.4 shows the locations of the different cities and Fig.5 shows the reference path for salesman to travel which is obtained by the nearest neighbor method C# is the language used for solving TSP

Fig.4 Locations of 50 cities Fig.5 Reference path obtained

The shortest distances and their corresponding paths are given in the Table.1

Table.1 Different paths and their shortest distance Algorithms No: of

Cities

Distance

1,33,46,6,40,15,20,3,17,11,38,23,35,28,19,10,13,14,45,49, 36,8,25,12,44,21,41,24,7,2,31,5,26,29,32,42,47,9,50,16,4,3 7,43,27,48,39,34,1

3595.22

1,6,40,15,46,33,20,18,17,22,38,23,45,49,36,8,25,12,44,21, 41,24,7,2,31,5,26,29,32,42,47,9,50,16,4,19,10,14,13,37,43, 27,48,39,34,3,11,28,1

3547.18

1,21,41,24,7,2,31,5,26,29,32,42,47,9,50,16,4,19,10,14,13,3 7,43,27,48,39,34,3,11,28,6,40,15,46,33,20,18,17,22,38,23, 45,49,36,8,25,12,44,1

3539.39

1,31,5,26,29,32,42,47,45,46,6,40,15,9,50,16,4,37,43,27,48, 39,34,33,20,3,17,11,38,23,35,28,19,10,13,14,41,24,7,2, 49, 36,8,25,12,44,21,1

3530.06

1,14,45,49,36,8,25,12,44,21,41,24,7,2,31,5,26,29,32,42,47, 33,46,6,40,15,20,3,17,11,38,23,35,28,19,10,13,43,27,48,39 ,34,50,16,4,37,9,1

3495.63

1,22,38,23,45,49,36,8,25,12,44,21,41,24,7,2,31,5,26,29,32, 42,47,9,50,16,4,19,10,14,13,37,43,27,48,39,34,3,11,28,6,4 0,15,46,33,20,18,17,1

3443.56

1,19,10,13,14,41,24,7,2,31,5,26,29,32,42,47,45,49,36,8,25, 12,44,21,9,50,16,4,37,43,27,48,39,34,33,46,6,40,15,20,3,1 7,11,38,23,35,28,30,1

3423.18

1,49,36,8,25,12,44,21,41,24,7,2,31,5,26,29,32,42,47,9,50,1 6,4,37,43,27,48,39,34,33,46,6,40,15,20,3,17,11,38,23,35,2 8,19,10,13,14,1

3241.8

1,21,9,50,16,4,37,43,27,48,39,34,33,46,6,40,15,20,3,17,11, 38,23,35,28,30,19,10,13,14,41,24,7,2,31,5,26,29,32,42,47, 45,49,36,8,25,12,44,1

3258.7

1,92,48,39,34,3,22,38,23,45,49,11,28,6,40,15,46,33,20,18, 17,50,16,4,19,10,14,13,37,36,8,25,12,44,21,41,24,7,2,31,5, 2643,29,32,42,47,1

3232.81

1,9,50,16,4,37,43,27,48,39,34,33,46,6,40,15,20,3,17,11,38, 23,35,28,19,10,13,14,41,24,7,2,31,5,26,29,32,42,47,45, 49, 36,8,25,12,44,21,1

3217.22

Thus, the best shortest distance is given in the Table 2

Table 2 Best shortest distance Algorithms No: of

Cities

Distance

Time Consumed

ABC

50

1,31,5,26,29,32,42,47,45,46,6,40,15,9,50 ,16,4,37,43,27,48,39,34,33,20,3,17,11,38 ,23,35,28,19,10,13,14,41,24,7,2,49,36,8, 25,12,44,21,1

1,19,10,13,14,41,24,7,2,31,5,26,29,32,42 ,47,45,49,36,8,25,12,44,21,9,50,16,4,37, 43,27,48,39,34,33,46,6,40,15,20,3,17,11, 38,23,35,28,30,1

1,9,50,16,4,37,43,27,48,39,34,33,46,6,40 ,15,20,3,17,11,38,23,35,28,19,10,13,14,4 1,24,7,2,31,5,26,29,32,42,47,45,49,36,8, 25,12,44,21,1

The graphs shown below mainly describe the influence of ABC variants by varying the number

of cities The results are depicted in Fig.6 and Fig.7 The results show that for 50 iterations if the number of cities is increased the corresponding shortest distance and the time taken also is increased Further analysis of the graphs in terms of shortest distance and time, the linear increment is very less for PS-ABC when compared to I-ABC and ABC algorithm However, from the results, it is clear that PS-ABC performs well when compared to I-ABC and ABC algorithm in finding the shortest distance within the minimum span of time A sample results are explained in detail in the above Table 1 and Table 2 where the number of cities are taken as 50 and also the number of iteration is taken as 50 From the above sample results it shows that when the number of cities is taken as 50 the PS-ABC performs better in finding the least distance within the small amount of time

Fig.6 Comparison of performance of 3 algorithms for n number of cities in 50 iteration

Fig 7 Comparison of performance of 3 algorithms for n number of cities in terms of time

Conclusion

The optimization algorithms are mainly used for solving the NP-hard optimization problems In this work the variants of ABC algorithm are presented aiming at minimizing the distance of the tour and also find the correspondent optimal path For obtaining the performance the original ABC algorithm is compared with the variants of ABC algorithm (I-ABC and PS-ABC) The results show that the PS-ABC performs well in finding the shortest distance within the minimum span of time when compared to I-ABC and ABC algorithms

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