A Novel Hybrid Ant Colony OptimizationApproach to Terminal Assignment Problem

Mahendra Prasad

Dr. Alok Singh(Associate Professor)

School of Computer & Information ScienceUniversity of Hyderabad

August 12, 2016

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

Content

⇒ Introduction⇒ TA Problem Definition.⇒ Vogel Approximation Method.⇒ Ant Colony Optimization.⇒ Local Search.⇒ Selection Procedure.⇒ Fitness Function.⇒ Hybrid Ant Colony Approach.⇒ Computational Result.⇒ Conclusion & Future Work.

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

Introduction

Terminal Assignment Problem is an NP-hard problem oftelecommunication networks.The main objective is to assign a given collection of terminalsto a given collection of concentrators.Each terminal is assigned to one concentrator.Aggregate weight of assigned terminals to the concentratorshould not exceed that concentrator capacity.Minimum fitness value indicates better assignment ofterminals to concentrators.Our approach is a combination of Vogel ApproximationMethod, an Ant Colony Optimization, and a local search.

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

TA Problem Definition

Problem InstanceTerminal t1, t2, ..., tnWeight w1,w2, ...,wnConcentrator c1, c2, ..., cmCapacities q1, q2, ..., qm

Feasible SolutionAssign each terminal to oneof the concentrators.

Aggregate weight of assignedterminals should not exceedthe concentrator capacity.

Objective FunctionFitness function→ Distance between assigned

terminals and concentrator.→ Balance network.→ penalisation

Minimize fitness value.

Optimal Solution

• Minimum fitness value.

S. Khuri and T. Chiu, Heuristic Algorithms for the Terminal Assignment Problem[4].

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

Simple Example of TA Problem

(a) Terminals dataTerminal Weight Coordinates

1 5 (54,28)2 4 (28,75)3 4 (84,44)4 2 (67,17)5 3 (90,41)6 1 (68,67)7 3 (24,79)8 4 (38,59)9 5 (27,86)10 4 (07,76)

(b) Concentrators DataConcentrator Capacity Coordinates

1 12 (19,76)2 14 (50,30)3 13 (23,79)

Table : Example of TA problem

1 2 3 4 5 6 7 8 9 102 1 2 2 2 3 3 1 3 1

Table : Solution Representation

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

Vogel Approximation Method for TA problem

Nmax = round( nm ) + 1

VAM Procedure→ Calculate the distance between each terminal and each concentrator.→ Calculate difference of two least distance concentrators.→ Select max difference terminal and assign least distance concentrator.→ Select next max difference distance and repeat until Nmax assignment on the

concentrator or concentrator capacity is exhausted and mark that concentrator.→ Marked concentrator is exempted from the further processes.→ Repeat the process until all terminals have been assigned.

S. Korukoglu and S. Balli, An Improved Vogels Approximation Method for the Transportation Problem[5].

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

Ant Colony Optimization

Artificial ant has been inspired by the behavior of real ant.The artificial pheromone are a kind of distributed numericinformation which is modified by the ants.ACO algorithms are finite size of artificial ants whichcollectively searches the better solution.It is used for finding a shortest path.A sequence of local moves find the shortest paths.

Marco Dorigo Gianni Di Caro, Luca M. Gambardella, Ant Algorithms for Discrete Optimization[3].

A.Singh,A.S.Baghel, New Metaheuristic Approaches for the Leaf Constrained Minimum Spanning Tree Problem[6].

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

Ant Colony OptimizationACO Equations

Probability equation

Pij =τij

n∑i=1

eji ∗ τij

where, Pij is ACO probability, τij ispheromone, eji is a binary coefficient.

Probability Tradeoff

Vij = total tj ∗ Pij

where, Vij is the trade-off parameter,total tj is the total number of terminalassigned to concentratorj .

Evaporation equation

τij = (1− ρ) ∗ τijwhere, ρ is the evaporation rate.

Pheromone update

τij = τij + eji ∗ k ∗1

distanceij

where k is a positive integer, distanceij isthe distance between terminali andconcentratorj .

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

Local Search

→ Local Search is used for improving the solution.→ Local Search is applied in this problem in two ways:

exchanging assignment of terminals between two distinctconcentrators.balancing assignment of terminals to concentrators.

→ Both the local searches stop when the first improving move isfound.

→ It is applied in every iteration.

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

Selection Procedure

concentrator tj =Vtj

m∑j=1

Vtj

→ Define range of each concentrator.→ Apply Roulette wheel process for selection.→ If TA problem constraint is not satisfied then directly apply

according to the highest value of Vtj or highest range.

Thomas Stutzle, Holger H. Hoos, MAX – MIN Ant System[7].

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

Fitness Function

Count assigned terminals

totalc =n∑

t=1

{1, if(c(t) = c)0, otherwise

Balance function

balc =

{10, if(totalc = Nmax )

20 ∗ |(Nmax − totalc)|

Euclidean distance between terminal and concentrator

distt,c(t) =√(tix − cjx )2 + (tiy − cjy )2

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

Fitness Functioncontinued...

Penalisation Equation

penalisation =

{0, if(Feasible)500, Otherwise

Fitness Equationfitness = 0.9 ∗

m∑c=1

balc + 0.1 ∗n∑

t=1distt,c(t) + Penalisation

E. M. Bernardino et. al., A HACO Algorithm for Solving the Terminal Assignment Problem[1].

E. M. Bernardino et. al., Ant Colonies to Assign Terminals to Concentrators[2].

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

HACA Algorithm

beginApply vogel method for initial solution.Forward fitness and assigned terminal.Initialize pheromone value.foreach iterations do

foreach ants doCalculate ACO probability.Calculate value of Vtj .Pheromone evaporate.Pheromone update.Select concentrator for each terminal.Apply Local Search.Calculate fitness.

endCompare fitness.Update better fitness.

endreturn fitness value.

end

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

Computational Result

Table : Best solution quality and run time of various approaches

Instance HACO NHACO-LS NHACO-TS HACABestF Ts BestF Ts BestF Ts BestF Ts

1 65.63 <1s 65.63 <1s 65.63 <1s 65.63 <1s2 134.65 <1s 134.65 <1s 134.65 <1s 134.65 <1s3 270.26 <1s 270.26 <1s 270.26 <1s 270.26 <1s4 286.89 <1s 286.89 <1s 286.89 <1s 286.89 <1s5 335.09 2s 335.09 <1s 335.09 <1s 335.09 <1s6 371.12 3s 371.12 1s 371.12 <1s 371.12 <1s7 401.21 4s 401.21 1s 401.21 1s 401.21 2s8 563.19 14s 563.19 7s 563.19 7s 563.19 5s9 642.83 25s 642.83 7s 642.83 7s 642.83 5s

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

Computational ResultContinued..

Table : Average solution quality and standard deviation of various approaches

Instance HACO NHACO-LS NHACO-TS HACAAvgF Std AvgF Std AvgF Std AvgF Std

1 65.63 0.00 65.63 0.00 65.63 0.00 65.63 0.002 134.65 0.00 134.65 0.00 134.65 0.00 134.65 0.003 270.32 0.06 270.32 0.06 270.26 0.00 270.26 0.004 286.91 0.04 286.91 0.04 286.89 0.00 286.89 0.005 335.11 0.03 335.11 0.03 335.09 0.00 335.09 0.006 371.55 0.17 371.55 0.17 371.24 0.09 371.16 0.047 401.61 0.15 401.61 0.15 401.34 0.12 401.42 0.118 563.55 0.16 563.55 0.16 563.35 0.07 563.26 0.059 643.67 0.38 643.67 0.38 643.23 0.11 642.91 0.09

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

Conclusion & Future Work

Vogel Approximation Method concept has been used first timeto solve TA problem.Computational result show the effectiveness of our approach.

Like Vogel Approximation Method, other methods availablein the operations research literature for solving transportationproblem/assignment problem may be tried for TA problem.

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

References

E. M. Bernardino, A. M. Bernardino, J. M. Sanchez-Perez, J. A. Gomez-Pulido, and M. A. Vega-Rodriguez.A hybrid ant colony optimization algorithm for solving the terminal assignment problem.International Joint Conference on Computational Intelligence, pages 144–151, 2009.

E. M. Bernardino, A. M. Bernardino, J. M. Sanchez-Perez, J. A. Gomez-Pulido, and M. A. Vega-Rodriguez.Ant colonies to assign terminals to concentrators.Springer, SCI 373:165–178, 2011.

M. D. G. D. Caro and L. M. Gambardella.Ant algorithms for discrete optimization.Artificial Life, 5(2):137–172, 1999.

S. Khuri and T. Chiu.Heuristic algorithms for the terminal assignment problem.ACM Symposium on Applied Computing, pages 247–251, 1997.

S. Korukoglu and S. Balli.An improved vogels approximation method for the transportation problem.Mathematical and Computational Applications, 16(2):370–381, 2011.

A. Singh and A. S. Baghel.New metaheuristic approaches for the leaf constrained minimum spanning tree problem.Asia Pacific Journal of Operational Research, 25(4):575–589, 2008.

T. Stutzle and H. H. Hoos.MAX-MIN ant system.Future Generation Computer Systems, 16:889–914, 2000.

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem

.

Thank you

Mahendra Prasad and Dr. Alok Singh A Novel Hybrid ACO Approach to TA Problem