Group search optimizatoin to solve tsp

Group Search Optimization to solve Traveling
Salesman Problem
M. A. H. Akhand, A. B. M. Junaed, Md. Forhad Hossain
Dept. of Computer Science and Engineering, Khulna University of Engineering & Technology,
Khulna, Bangladesh
akhand@cse.kuet.ac.bd, abm.junaed@gmail.com, forhad.csekuet@yahoo.com

K. Murase
Dept. Human and Artificial Intelligent Systems, University of Fukui, Fukui, Japan

M. A. H. Akhand A. B. M. Junaed Md. Forhad Hossain

We are going to discuss….
Here, We are going to discuss about the following things
1.Traveling Salesman Problem (TSP)
2.Group Search Optimization (GSO)
3.Motivation of Solving TSP using GSO and
4.Group Search Optimization Algorithm (GSOA)
5.A comparative study of solving TSP using GSO algorithms and Other
Nature Inspired Algorithms (NIAs) and


Traveling Salesman Problems (TSP)
TSP is a problem of finding a least-cost sequence of cities where Each
city will be visited exactly once and the beginning and the ending city
will be the same.

Tour = D – C – B – A – D
Path Cost = 12 + 30 + 20 + 35 = 97


Group Search Optimization (GSO)
• A new kind of Computational Intelligence!!!
• Depending of Collective behavior of animals???
• Yes!! We see an animal to find or attempt to find resources such as
food, mates, oviposition, or nesting sites.


• The ultimate success of an animal’s
searching depends on
1.The strategies it uses in relationship to
the available of resources and their
spatial and temporal distributions in the
environment.
2.Its efficiency in locating resources and
3.The ability of species to adapt to long-
term or even short-term environmental
changes and the ability of an individual
to respond.



• Depending on these behavior of
animals, a novel optimization
algorithm has been proposed in 2009
called Group Search Optimization
(GSO) which was inspired by the
animal behavior, especially animal
searching (foraging) behavior.


Motivation of solving TSP using GSO
Motivation of Traveling Salesman Problem are mainly discussed in
three categories.
1.Biological Motivation
2.Engineering Motivation and
3.Real-life Motivation


Biological Motivation
• Animals normally search for food in group. They get benefited sharing
information among themselves.
• Animals are mainly two types: Producer & Scrounger.
• A model named Producer-Scrounger (PS) Model has been developed
from these two types of animals.
• On the basis of these behavior, GSO algorithm to solve TSP has been
developed.


Engineering Motivation
• Using GSO, Engineers can easily solve Traveling Salesman Problem.
• A new era of Engineering has been opened after the invention of GSO
algorithms to solve TSP.
• Engineers will get the opportunity to research on this algorithm and
they will try to increase the optimality of this algorithms.
• Algorithm of GSO to solve TSP gives better results than some other
Nature Inspired Algorithms.


Real-Life Motivation
TSP can be used in various sector in the real world and here we can use
GSO to find the optimum solutions,
•Arranging School bus routes
•Merrill Flood, one of the pioneers of TSP research in the 1940s.
•Transportation of farming equipment from one location to another
location.
•More recent applications involve the scheduling of service calls at
cable firms, the delivery of meals to homebound persons, the
scheduling of stacker cranes in warehouses, the routing of trucks for
parcel post pickup, and a host of others.
•Scheduling of a machine to drill holes in a circuit board or other object


Group Search Optimization (GSO) to solve TSP
A. GSO to solve TSP:
1. Randomly initialize tour for all the members (Xi) for N cities and calculate
fitness values (i.e., f(Xi)) of each.
2. While (the termination conditions are not met) {
3. For (each members i in the group) {
3.a. Perform producing:
•Find the producer XP of the group.
•Select a city (C) randomly.
•Select top 10% nearest cities from C according to distance. Let these cities
are [N1,N2…Nx]
•Now create new tours (X’P) using these cities.
•Producer will fly to X’P (i.e., XP= X’P) if f(X’P) is better than f(XP).


3.b. Perform scrounging:
i.Select a number of group members (normally 80% of the members) as
scroungers.
ii.Generate SS for each scrounger using Eq. S’ = S + lSS =l( P – S) and
move it towards the producer using the SS.
3.c. Perform dispersion:
i.Select rest of the members as dispersed.
ii.Randomly generate a SS for each dispersed member and fly to new
tour.
3.d. Calculate the fitness value of current members: f(Xi)
} // End For
} // End While

• GSO a is population based optimization technique on the metaphor of
producer-scrounger based social behavior of animals.
• GSO has been found as an efficient method for solving function
optimization problems for which it modeled.
• In this study we employ the concept of Swap Operator (SO) and Swap
Sequence (SS) to modify GSO for TSP.
• The modified GSO (mGSO) was tested on a number of benchmark
TSPs and results compared with some existing approaches.


Swap Operator and Swap Sequence have been used.
A. Swap Operator (SO): A Swap Operator (SO) swaps two cities in a
tour indicated in the SO,
Suppose, a TSP problem has 6 cities and a solution is 1-2-3-6-4-5. A SO
(2, 3) gives the new solution S’,
S’=S+SO(2,3)=(1-2-3-6-4-5) + SO(2,3) = 1-3-2-6-4-5 . Here ‘+’ means
to apply SO(s) on the solution.
B. Swap Sequence (SS): A swap sequence (SS) is made up of one or
more swap operators. SS = (SO1 , SO2 , SO3 ,SO4,…,. SOn) where
SO1, SO2, SO3, SO4 …, SOn are the swap operators.


C. Construction of Swap Sequence: Suppose Two solutions A and B.
For A(1-2-3-4-5) and B(2-3-1-5-4),
A(1) = B(3) = 1.
So the first Swap Operator is SO1(1,3).
B1 = B + SO(1,3) = (1-3-2-5-4)
Now A(2) = B(3) = 2. So second operator is SO2(2,3).
Applying in this way, we get a SS
SS= A-B = ( SO(1,3), SO(2,3), SO(4,5) )


D. Producer Scanning for TSP:
•The producers use local search method and select top 10% nearest
cities according to distance.
•Let, one of the nearest cities is N1. Now the producer will create
connection between these two cities.
•It will put C before N1 and make a connection between these two
cities. Hence it will get a new tour.
•Then it will put C after N1 and will get another tour. Then put N1
before C and after C and hence get 2 new tours.


Suppose the tour of producer is 1-2-3-4-6-5-7-8 and the randomly
selected city is 6. Let, one of these nearest cities is 7.
So according to the description above, we will have four new tours.
These are:
a.1-2-3-4-6-5-7-8
b.b. 1-2-3-4-5-7-6-8
c. 1-2-3-4-7-6-5-8
d. 1-2-3-4-6-7-5-8
If another nearest city is 5, then we will have only one new tour, since
there is a direct connection already exists between 6 and 5.
a.If producer find better tour than its current one, it will conceive the
new best tour. Producer will be d (1-2-3-4-6-7-5-8) if its cost is less
than current position.

E. Scrounging and Dispersion for TSP
•80% of the members are scroungers
•Rest of the members will be dispersed from the group [14]
•Scrounger(s) moves to Producer(P) using SS
•Portion of SS will apply on S to get the new tour
S’ = S + lSS =l( P – S)
•Dispersed members will move to new tours based on randomly
generated SSs


Comparative Study
A. Benchmark Problems and Experimental Setup :
•15 benchmark problems from TSPLIB [15] where number of cities
varied from 14 to 100. For example, burma14 has 14 cities.
•A city is represented as a coordinate in a problem. Therefore the cost is
found after calculating distance using the coordinates.
•For proper understanding, we also solved the benchmark problems
with Genetic Algorithm (GA) [9-10], Ant Colony Optimization (ACO)
[12] and Particle Swarm Optimization (PSO) [6].
•The algorithms are implemented on Visual C++ of Visual Studio 2010.
•The experiments have been done on a PC (Intel Core 2 Duo E7200
@2.53GHz CPU, 1GB RAM) with Windows 7 OS.


Comparative Study
•For the fair comparison, number of generation was 500 for the
algorithms.
•The population size was 50 for GA, PSO and mGSO, equal to number
cities in ACO
•For GA, tournament selection was used and both crossover and
mutation rates are 10%.
•selected parameters are not optimal values, but selected for simplicity as well
as for fairness in observation.
•In ACO, alpha is set to 1 and beta is set to 3.

B. Experimental Results: Here, we are going to compare the
experimental results among themselves.

Comparative Study Table 1
Average Tour Cost of 30 Runs
Problem
GA ACO PSO mGSO
burma14 30.87 31.21 31.47 31.05
ulysses16 74.08 77.13 74.57 74.25
ulysses22 79.04 86.9 81.53 77.6
fri26 710.39 646.48 738.04 678.45
bayg29 9247.92 9964.78 10846.49 9774.3
bays29 9743.58 9964.78 10750.24 9748.92
att48 45083.24 39513.68 49693.59 38603.51
eil51 529.45 435.71 590.27 476.67
berlin52 10469.52 8072.06 11300.24 8761.45
st70 1062.43 734.19 1281.67 854.61
eil76 712.6 602.95 960.26 634.45
pr76 161734 127371.7 214716.1 129940.1
gr96 899.88 594.83 1223.41 617.05
rat99 1995.17 1369.53 2847.98 1467.34
kroB100 37796.45 25894.32 58173.24 30317.55
Average 18677.91 15024.02 24220.61 15470.49
Best/Worst 4/0 9/2 0/13 2/0

Comparative Study
Table 1 description
•ACO is found best for nine cases and achieved best average tour cost.
•ACO is shown worst for two cases.
•Proposed mGSO is shown competitive result to ACO showing worst
for no one.
•At a glance mGSO seems competitive to ACO and outperforms GA
and PSO for the average result presented in the Table I.


Comparative Study Table 2
Best (i.e., Minimum) Tour Cost from 30 Runs
Problem
GA ACO PSO mGSO
burma14 30.87 31.21 30.87 30.87
ulysses16 74 77.13 73.99 73.99
ulysses22 78.98 86.9 75.31 75.31
fri26 678.33 646.48 639.17 635.58
bayg29 9213.9 9964.78 9787.8 9076.98
bays29 9456.78 9964.78 9323.12 9074.15
att48 44351.03 38989.37 40822.94 34762.09
eil51 505.08 435.71 540.47 422.89
berlin52 10243.93 8046.06 9811.75 8076.23
st70 1022.31 734.19 1138.96 714.26
eil76 683.5 602.4 877.62 585.91
pr76 153133.1 127371.7 183023.4 119128.4
gr96 866.82 594.83 1069.4 540.39
rat99 1885.1 1369.53 2435.5 1361.6
kroB100 33048.27 25792.4 51493.34 25550.55
Average 17684.8 14980.5 20742.91 14007.28
Best/Worst 1/3 1/5 3/7 14/0

Comparative Study
Table 2 description
•mGSO is shown to achieve the lowest average tour cost of 14007.28.
•On the other hand the values for GA, ACO and PSO were 17684.8,
14980.5 and 20742.91, respectively.
•On the basis of best/worst summary, mGSO is shown to achieve best
tour with shortest path for 14 cases out of 15 cases.


Comparative Study
• Considering Table I and Table II, mGSO is better than ACO in case of
best of the runs although it is inferior to ACO for average results.
• ACO uses population sizes as the number of cities. - -
• Therefore, problem having large number cities (more than 50), ACO got
benefit of larger population size whereas the population size was fixed 50 for
mGSO for such problems.
• Therefore, ACO outperformed mGSO and others (GA and PSO) for large
problems as it is seen in the Table I.
• On the other hand, ACO are unable to work with population size larger than
number of cities that make it inferior to any other methods for small
problems.
• Considering problems having various sizes mGSO is the best suitable
algorithm

Comparative Study
Population size = 50 (fixed except ACO)
Number of Generation = 10 to 1000

Figure 1. Tour Cost vs Generation fixing population size at 100.


Comparative Study
• Figure 1 compares the tour cost varying generation from 10 to 1000
fixing population size at 50
• ACO is almost invariant with respect to generation showing worse
performance.
• GA, PSO and mGSO are found to improve up to 100 generations and
after that they were also invariant.
• However, mGSO is shown to achieve better performance than others.


Comparative Study
Population size = 10 to 500 (except ACO)
Number of generation = 50 (fixed)

Figure 3. Tour Cost vs Population Size fixing Generation at 500


Comparative Study
• Figure 2 compares the tour cost varying population size from 10 to
500 fixing generation at 500.
• Population size enlargement helps to improve performance ACO in
the initial stage because population size larger than number of cities
might not effective for ACO.
• On the other hand although GA, PSO and ACO have shown better
than mGSO for small population size, mGSO is shown to improve its
performance better than others when population increases and
outperformed them for larger population size, e.g., more than 300.
• Therefore it is good for mGSO to improve performance working with
larger population size.


Comparative Study
• The proposed modified GSO (mGSO) tested on a large number of
benchmark TSPs and is compared with some other popular algorithms
such as
• GA, ACO and PSO.
• mGSO is shown to achieve best results (i.e., tours with shortest path
costs) for several problems and other cases it was highly competitive.


References
References:
[1] R. Matai, S. P. Singh and M. L. Mittal, “Traveling Salesman
Problem:An Overview of Applications, Formulations, and Solution
Approaches,” Traveling Salesman Problem, Theory and Applications,
Edited by D. Davendra, InTech, pp 1-24, 2010.
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[3] D. Whitely, “A genetic algorithm tutorial,” Statistics and
Computing4, pp. 65-85,1994.
[4] E. Bonabeau, M. Dorigo, G. Theraulaz, Swarm Intelligence: From
Natural to Artificial Systems, Oxford University Press, Oxford, 1999.


References
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optimization metaheuristic: basis, models and new trends, Mathware
and Soft Computing 9, pp 141-175, 2002.
[6] R. Eberhart, J. Kennedy. “A New Optimizer Using Particles Swarm
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[8] L. Wong, M. Y. H. Low and C. S. Chong, “A Bee Colony
Optimization Algorithm for Traveling Salesman Problem,” Second
Asia International Conference on Modeling & Simulation, no. 978-0-
7695-3136-6/08, 2008. IEEE DOI 10.1109/AMS.2008

References
[9] J. Krause and G. D. Ruxton, Living in Groups. Oxford Series in
Ecology and Evolution. Oxford University Press, 2002.
[10] C. W. Clark and M. Mangel, “Foraging and flocking strategies:
Information in an uncertain environment,” Amer. Naturalist, vol. 123,
pp. 626–641, 1984.
[11] C. J. Barnard and R. M. Sibly, “Producers and scroungers: A
general model and its application to captive flocks of house
sparrows,”Animal Behavior, vol. 29, pp. 543–550, 1981.
[12] L. A. Giraldeau and G. Beauchamp, “Food exploitation: Searching
for the optimal joining policy,” Trends Ecology & Evolution, vol. 14,
no. 3, pp. 102–106, 1999.


References
[13] S. He, Q. H. Wu and J. R. Saunders, “A novel group search
optimizer inspired by animal Behavioral ecology,” in Proc. 2006 IEEE
Congr. Evol. Comput., Vancouver, BC: Sheraton Vancouver Wall
Center, pp. 1272–1278, Jul. 2006.
[14] S. He, Q. H. Wu, and J. R. Saunders, “Group Search Optimizer: An
Optimization Algorithm Inspired by Animal Searching Behavior,” IEEE
Transactions On Evolutionary Computation, vol. 13, no 5, pp. 973-990,
October 2009.
[15] TSPLIB - A library of sample instances for the TSP. Available:
http://www.iwr.uni-heidelberg.de/groups/ comopt
/software/TSPLIB95/tsp


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