Group search optimizatoin to solve tsp

M. A. H. Akhand, A. B. M. Junaed, Md. Forhad HossainDept. of Computer Science and Engineering, Khulna University of Engineering & Technology,

Khulna, Bangladesh

[email protected], [email protected], [email protected]

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

Group Search Optimization to solve Traveling Salesman Problem

Md. Forhad HossainA. B. M. JunaedM. A. H. Akhand

mailto:[email protected]



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

We are going to discuss….


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.

Traveling Salesman Problems (TSP)


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

• 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.

Group Search Optimization (GSO)


• 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 Traveling Salesman Problem are mainly discussed in three categories.

1.Biological Motivation

2.Engineering Motivation and

3.Real-life Motivation

Motivation of solving TSP using GSO


• 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.

Biological 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.

Engineering 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

Real-Life Motivation


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).

Group Search Optimization (GSO) to solve TSP


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.



something

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



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


Comparative Study Table 1


Problem Average Tour Cost of 30 RunsGA 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

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


Comparative Study Table 2


ProblemBest (i.e., Minimum) Tour Cost from 30 Runs

GA ACO PSO mGSOburma14 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

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

Comparative Study


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

• 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)

Comparative Study


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

• 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.

Comparative Study


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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[4] E. Bonabeau, M. Dorigo, G. Theraulaz, Swarm Intelligence: From Natural to Artificial Systems, Oxford University Press, Oxford, 1999.


References

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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

References
