Hybrid Method for Terminal Assignment Problem

Ant Colony Optimization Technique for TerminalAssignment Problem

Mahendra Prasad14MCMI17

Under the guidance ofDr. Alok Singh

School of Computer & Information ScienceUniversity of Hyderabad

Aug. 4, 2016Mahendra Prasad Ant Colony Optimization Technique for TA Problem

Content

⇒ Introduction⇒ TA Problem Definition.⇒ Important Existing Approaches.⇒ Fitness Function.⇒ Ant Colony Optimization method.⇒ Modified Vogel Approximation Method.⇒ Hybrid Ant Colony Approach.⇒ Computational Result.⇒ Conclusion & Future Work.

Mahendra Prasad Ant Colony Optimization Technique for 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 the capacity of concentrator.Minimum fitness value indicates better assignment.


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].


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


Simple Example of TA Problem


Important Existing Approaches

1. Local Search Genetic Algorithm (LSGA).2. Genetic Algorithm with Multiple Operator (GAMO).3. Tabu Search (TS).4. Bees Algorithm (ABC).5. Hybrid Differential Evoluation (HDE).6. Multiple Strategy HDE (MHDE).7. Discrete Differential Evoluation (DDE).8. Hybrid Ant Colony Optimization (HACO).9. New Improved HACO (NHACO).10. Discrete Particle Swarm Optimization (DPSO).11. Hybrid Population Based Incremental Learning (HPBIL).12. New HACO with Local Search (NHACO-LS).13. New HACO with Tabu Search (NHACO-TS).

E. M. Bernardino, A. M. Bernardino, J. M. Sanchez-Perez, J. A. Gomez-Pulido and M. A. Vega-Rodriguez.


Fitness Function

Count assigned terminals

totalc =n∑

t=1

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

0, otherwise

Balance function

balc =

{10, if(totalc = round( n

m ) + 1)

20 ∗ |(round( nm ) + 1− totalc)|

Euclidean distance between terminal and concentrator

distt,c(t) =√

(tix − cjx )2 + (tiy − cjy )2


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].


Ant Colony Optimization (ACO)

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.


TA Problem using ACO

ACO Probability

pij(t + 1) =

ταij (t)η

βij (t)∑

g∈N ji

ταij (t)ηβij (t), if g ∈ N j

i

0, if g /∈ N ji

where,→ N j

i Number of terminals connected to j th concentrator.→ ηij = 1

distanceij→ distanceij is euclidean distance terminali to concentratorj .→ τ is pheromone.→ α and β are constant.


TA Problem using ACOContinued...

Delta tau value

∆τ =1

(10 + C(itbest)− C(best))

where, C(itbest) is the best solution of iteration and C(best) is bestsolution found since the begining.

Pheromone Evaporationτij = (1− ρ)τijwhere,→ ρ evaporator operator.→ ρ ∈ (0 ≤ ρ < 1)

Pheromone updateτij(t + 1) = τij(t) + ∆τij(t)where,→ t is current time.→ τ is pheromone.

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].


Selection Procedure

Pseudo Random Proportional Rule

This rule has been used for concentrator selection.Every iteration progressively builds a better selection.

Concentratorj =

{maxij{ταij ∗ η

βij } if x0 ≤ x

RouletteWheelProcess otherwise

where,→ x0 is a constant taken in the interval [0,10].→ x is a random variable uniformly distributed in [0,10].

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


Selection ProcedureContinued...

Roulette Wheel process

This is probability based selectionprocedure.Using this rule, a random number generateevery time.Higher probability value concentrator ismore probable to select.

Concentratorj =Pij

m∑y=1

Pij

∗ 100

where,→ m is total number of concentrator.→ Pij is ACO probability of terminali to concentratorj .

Example


ACO Algorithm

beginRead a given data.Calculate Euclidean distance.Initialize pheromone value.foreach iterations do

foreach ants doCalculate ACO probability.Pheromone evaporate.Pheromone update.Select concentrator.

endCalculate fitness.Update better fitness.

endreturn fitness value.

end


Computational Result of ACO

P NHACO-TS ACOBestF Ts AvgF Std BestF Ts AvgF Std

1 65.63 <1s 65.63 0.00 65.63 <1s 65.63 0.002 134.65 <1s 134.65 0.00 134.65 <1s 134.65 0.003 270.26 <1s 270.26 0.00 278.65 <1s 281.09 1.434 286.89 <1s 286.89 0.00 297.23 <1s 298.90 1.665 335.09 <1s 335.09 0.00 338.79 1s 338.79 0.006 371.12 <1s 371.24 0.09 386.40 1s 389.29 1.597 401.21 1s 401.34 0.12 417.09 3s 421.13 2.008 563.19 7s 563.35 0.07 581.46 7s 581.46 0.009 642.83 7s 643.23 0.11 662.38 7s 662.38 0.00

Given Input ValueNumber of ants=10, Number of iterations=100Initial Pheromone Value τ=100, α=1, β=2, ρ=0.01


Modified Vogel Approximation Method (MVAM)

MVAM is a combination of the following techniques:→ Vogel Approximation Method→ Local Search


Vogel Approximation Method (VAM)

Vogel Approximation Method

Compute difference between twolowest distances.Select max difference terminal.Assign lowest distanceconcentrator.Exempt the already assignedconcentrator from the repeatingprocess.Repeat the process until allterminals have been assigned.

Simple Example

c1 c2 c3 c4t1 5 9 12 18t2 10 31 35 9t3 25 29 12 4t4 16 24 6 27

Two least distance diff.t1 t2 t3 t44 1 8 10

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





Simple Example

c1 c2 c3 c4t1 5 9 18t2 10 31 9t3 25 29 4

Two least distance diff.t1 t2 t34 1 21






Simple Example

c1 c2 c3 c4t1 5 9t2 10 31

Two least distance diff.t1 t24 21






Simple Example

c1 c2 c3 c4t1 5 9 12 18t2 10 31 35 9t3 25 29 12 4t4 16 24 6 27

t1 → c2, t2 → c1,t3 → c4, t4 → c3



VAM Approach 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.


Local Search

→ Local Search is used for improving the solution.→ Local Search is applied 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.


MVAM ApproachSelection Procedure

concentrator tj =1

distancetj

Selection procedure

→ Define a range for all concentrators.→ Select a concentrator using roulette wheel process.→ If the selected concentrator does not satisfy the constraint then

directly select according to the highest range concentrator.


MVAM Algorithm

beginApply vogel method for initial solution.Forward fitness and assigned terminals.foreach iterations do

initialize p to zero.while p < vloop do

Apply Local Search.Select concentrator for each terminal.increment value of p by one.Calculate fitness.

endCompare fitness.Update better fitness.


end


Computational Result of MVAM

P NHACO-TS MVAMBestF Ts AvgF Std BestF Ts AvgF Std

1 65.63 <1s 65.63 0.00 65.63 <1s 65.63 0.002 134.65 <1s 134.65 0.00 134.65 <1s 134.65 0.003 270.26 <1s 270.26 0.00 271.07 <1s 271.07 0.004 286.89 <1s 286.89 0.00 288.07 <1s 288.07 0.005 335.09 <1s 335.09 0.00 335.88 <1s 335.88 0.006 371.12 <1s 371.24 0.09 372.34 <1s 372.34 0.007 401.21 1s 401.34 0.12 404.25 <1s 404.25 0.008 563.19 7s 563.35 0.07 563.83 2s 563.83 0.009 642.83 7s 643.23 0.11 644.48 2s 644.48 0.00

Given Input ValueNumber of iterations=1000, vloop value is 4


Hybrid Ant Colony Approach (HACA)

HACA is a combination of the following techniques:→ Ant Colony Optimization→ Vogel Approximation Method→ Local Search


Hybrid Ant Colony ApproachACO 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 .


Hybrid Ant Colony ApproachSelection Procedure

concentrator tj =Vtj

m∑j=1

Vtj

Similar to the other approaches, the selection procedure is samebut the equation is different.→ 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.


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.


end


Computational Result of HACA

P NHACO-TS HACABestF Ts AvgF Std BestF Ts AvgF Std

1 65.63 <1s 65.63 0.00 65.63 <1s 65.63 0.002 134.65 <1s 134.65 0.00 134.65 <1s 134.65 0.003 270.26 <1s 270.26 0.00 270.26 <1s 270.26 0.004 286.89 <1s 286.89 0.00 286.89 <1s 286.89 0.005 335.09 <1s 335.09 0.00 335.09 <1s 335.09 0.006 371.12 <1s 371.24 0.09 371.12 <1s 371.16 0.047 401.21 1s 401.34 0.12 401.21 2s 401.42 0.118 563.19 7s 563.35 0.07 563.19 5s 563.26 0.059 642.83 7s 643.23 0.11 642.83 5s 642.91 0.09

Given Input ValueNumber of ants=10, Number of iterations=200, Initial Pheromone(τ)=100,Evaporation rate(ρ)=0.01, Constant Value ‘k’=2


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.


.

This dissertation work has been accepted as a research papertitled A Novel Hybrid Ant Colony Optimization Approachto Terminal Assignment Problem in the ACM InternationalConference on Advances in Information Communication Tech-nology & Computing (AICPS-01176/AICTC2016).


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.


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Hybrid Method for Terminal Assignment Problem

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