ant colony optimization technique for terminal assignment problem

Ant Colony Optimization Techniquefor Terminal Assignment Problem

A dissertation submitted to the University of Hyderabad inpartial fulfilment of the requirements for the award of degree of

Master of Technologyin

Artificial Intelligenceby

Mahendra Prasad14MCMI17

School of Computer and Information SciencesUniversity of Hyderabad

Hyderabad-500046

June 2016

CERTIFICATE

This is to certify that the dissertation entitled Ant Colony Op-timization Technique for Terminal Assignment Problem submittedby Mahendra Prasad, bearing Reg. No. 14MCMI17, in partial fulfilmentof the requirements for the award of Master of Technology in Artificial In-telligence is a bonafide work carried out by him under my supervision andguidance. The dissertation has not been submitted previously in part or infull to this or any other University or Institution for the award of any degreeor diploma.

Dr. Alok SinghSupervisor

School of Computer & Information Sc.University of Hyderabad

DeanSchool of Computer & Information Sc.

University of Hyderabad

Declaration

I, Mahendra Prasad, hereby declare that this dissertation entitled AntColony Optimization Technique for Terminal Assignment Problemsubmitted by me under the guidance and supervision of Dr. Alok Singh, is abonafide work carried out by myself. I also declare that this dissertation hasnot been submitted previously in part or in full to this University or otherUniversity or Institution for the award of any degree or diploma. I herebyagree that my dissertation can be deposited in Shodganga/INFLIBNET. Areport of plagiarism statistics from the University Librarian is enclosed.

Date:Place:

Mahendra Prasad14MCMI17

//Countersigned//

Dr. Alok Singh(Signature)

Acknowledgement

It gives me immense pleasure to express my deepest sense of gratitude andsincere thanks to our highly respected and esteemed supervisor Dr. AlokSingh, for their valuable guidance, encouragement and help for completingthis dissertation. Their useful suggestions, fruitful discussions on design andimplementation of this dissertation and their valuable guidance is sincerelyacknowledged.

I am grateful to Prof. Arun Agarwal, Dean of SCIS, for providingexcellent computing facilities and a congenial atmosphere for progressingwith my project.

My thanks and appreciations also goes to my lab colleagues in developingthe dissertation and having valuable discussions and inputs.

Last but not the least I would like to express my sincere thanks to all myfriends and others who have helped directly or indirectly during this disser-tation work.

Mahendra Prasad(14MCMI17)

Abstract

Terminal Assignment (TA) problem is a well-known problem in the designof telecommunication networks. It is an NP-hard combinatorial optimiza-tion problem. This problem is regarding assignment of a given collection ofterminals to a given collection of concentrators subject to some constraints.Each terminal is assumed to have a positive weight and each concentrator isassumed to have fixed capacity. Terminal Assignment is to assign terminalsto concentrators in such a way that each terminal is assigned to exactly oneconcentrator and aggregate weight of terminals assigned to the concentratorshould not exceed that concentrator capacity. The objective of TA Problemconsidered by us is to minimize the weighted sum of distances of terminalsfrom their assigned cocentrators and unbalancing in loads on different con-centrators as measured by a balance function.

Ant Colony Optimization (ACO) algorithm for solving combinatorial op-timization problems is based on intelligent foraging behavior of colony ofants. In telecommunication networks, a large number of optimization prob-lems have been solved by using the ACO algorithms. ACO is a populationbased probabilistic metaheuristic technique. Recent research has focused onthe development of hybrid ACO variants by combining ACO with problemspecific heuristics. These hybrid variants achieve better performance in com-parison to non hybrid varients.

We have developed three methods for TA poblem which are Ant ColonyOptimization, Modified Vogel Approximation Method, and an hybrid ACOalgorithm with the aim of achieving results as good as or better than previ-ously proposed approaches. Our Hybrid ACO for TA problem approach isa combination of Vogel approximation method, an ant colony optimizationalgorithm and a local search. Computational results on standard benchmarkinstances of TA problem show that the performance of our hybrid approachis comparable with other state-of-the-art metaheuristic approaches for TAproblem.

Contents

1 Terminal Assignment Problem 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The TA Problem . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Problem instance . . . . . . . . . . . . . . . . . . . . . 21.2.2 Feasible solution . . . . . . . . . . . . . . . . . . . . . 21.2.3 Objective function . . . . . . . . . . . . . . . . . . . . 31.2.4 Optimal solution . . . . . . . . . . . . . . . . . . . . . 3

1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Greedy Algorithms for TA problem . . . . . . . . . . . . . . . 51.5 Greedy Anylysis . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Ant Colony Optimization 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 ACO Metaheuristic . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Behavior of Ants . . . . . . . . . . . . . . . . . . . . . 82.2.2 Artificial Pheromone . . . . . . . . . . . . . . . . . . . 92.2.3 Simple Ant Colony Optimization . . . . . . . . . . . . 10

2.3 ACO Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 ACO for Terminal Assignment Problem 143.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 ACO Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Basic steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Fitness Function . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Selection Process . . . . . . . . . . . . . . . . . . . . . . . . . 19

i

3.5.1 Pseudo Random Proportional Rule . . . . . . . . . . . 193.5.2 Roulette Wheel Process . . . . . . . . . . . . . . . . . 20

3.6 Boosting Process . . . . . . . . . . . . . . . . . . . . . . . . . 213.7 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.8 Computational Result . . . . . . . . . . . . . . . . . . . . . . 22

4 Modified Vogel Approximation Method 254.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Vogel Approximation Method to Solve Terminal Assignment

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.1 Selection Procedure . . . . . . . . . . . . . . . . . . . . 274.2.2 Evaluation Function . . . . . . . . . . . . . . . . . . . 284.2.3 MVAM Algorithm . . . . . . . . . . . . . . . . . . . . 29

4.3 Computational Result . . . . . . . . . . . . . . . . . . . . . . 29

5 A Hybrid Ant Colony Optimization Approach 325.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2 Important ACO Equations . . . . . . . . . . . . . . . . . . . . 33

5.2.1 Local Search . . . . . . . . . . . . . . . . . . . . . . . . 345.2.2 Selection Procedure . . . . . . . . . . . . . . . . . . . . 355.2.3 Hybrid Ant colony Algorithm . . . . . . . . . . . . . . 36

5.3 Computational Result . . . . . . . . . . . . . . . . . . . . . . 365.3.1 Comparison to the previous approaches . . . . . . . . . 385.3.2 Result Analysis . . . . . . . . . . . . . . . . . . . . . . 39

6 Conclusion and Future Work 40

Bibliography 41

ii

List of Figures

1.1 A simple example of TA problem . . . . . . . . . . . . . . . . 5

2.1 Shortest Path Selection by Ants[13] . . . . . . . . . . . . . . . 92.2 Ants finding shortest path . . . . . . . . . . . . . . . . . . . . 10

3.1 Eaxmple of Roulette Wheel Process . . . . . . . . . . . . . . . 203.2 Concentrator selection probability . . . . . . . . . . . . . . . . 21

iii

List of Tables

1.1 Example of TA problem . . . . . . . . . . . . . . . . . . . . . 41.2 Terminal representation with assigned concentrator . . . . . . 4

3.1 Results - Compare Bets Fitnesses and Execution Time . . . . 233.2 Comparison of Average Fitnesses and Standard deviations . . 24

4.1 Best Fitnesses and Time Execution compare to different ap-proaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Average Fitnesses and Standard deviations compare to differ-ent approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.1 Best solution quality and run time of various approaches . . . 375.2 Average solution quality and standard deviation of various ap-

proaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

iv

Chapter 1

Terminal Assignment Problem

1.1 Introduction

Terminal Assignment Problem is a combinatorial optimization problem forthe telecommunications network. The primary aim is to minimize the dis-tance and the fitness. For a given sets of terminals and concentrators, theterminals have some weights specified, and concentrators have some specifiedcapacity. The capacity of by the concentrator and the weights of terminalsmay vary. The location of the terminals and the concentrators are fixed andgiven in the coordinates form[14]. In TA problem assigning terminals to theconcentrator should constitute three main constraints.

• One terminal assigned to only one concentrator. More than one termi-nals can be assign to the one concentrator.

• Aggregate weight of terminals which are assigned to the same concen-trator should not exceed the capacity of the concentrator.

• Assignment of the terminals to concentrators should be balanced.

In TA problem the terminals are connected to the concentrator and the con-centrators are connected to the central equipment of the telecommunicationsnetwork. The distance between the terminal to the concentrator is calculateusing the Euclidean distance. The objective of the TA Problem is minimizing

1

CHAPTER 1. TERMINAL ASSIGNMENT PROBLEM

distance and balance distribution among the concentrators.

distanceij =√

(txi − cxj)2 + (tyi − cyj)2 (1.1)

where, x and y are given coordinates. distanceij is distance from terminali

to concentratorj.

1.2 The TA Problem

The following is a formal definition of the terminal assignment problem, forcombinatorial optimization problems[21]. We introduce concept and nota-tions that we use in subsequent subsections of this work.

1.2.1 Problem instance

Terminal Assignment Problem is described as

• Total number of n distinct terminals t1, t2, ..., tn.

• Total number of m distinct concentrators c1, c2, ..., cm.

• Weight or capacity of terminals w1, w2, ..., wn.

• Capacity of concentrators p1, p2, ..., pm.

wi is the weight, or capacity requirement of the terminal ti. The weightsand capacity are positive integers and wi < minimum(p1, p2, ...., pc) fori=1,2,....,t. The ‘t’ terminals and ‘c’ concentrators are placed on the Eu-clidean distance equation, i.e, ti has coordinates (tix, tiy) and cj is located at(cjx, cjy) where x and y are coordinates of both(terminals and concentrators).

1.2.2 Feasible solution

Assign each terminal to one of the concentrators such that aggregate weightsof assigned terminals to the concentrator does not exceeds concentrator ca-pacity. In other words, a feasible solution to the terminal assignment problem

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is:A vector ~v = v1, v2, ......., vt where vi=j means that 1 ≤ vi ≤ pc. where, viis an integer, i=1,2,....,t. Every terminals assigned to the concentrator andsatisfy constraint which is

∑i∈nj

wi ≤ pj, where j=1,2,.......,c (i.e, means that

aggregate weight of assigned terminals to the concentrator capacity does notexceeded) where nj = {i|vi = j}; i.e, nj represents that number of terminalsthose are assigned to concentrator j.

1.2.3 Objective function

An Objective of this problem minimizing the assigned terminals distanceto concentrator. For the purpose of minimizing used euclidean distanceequation(1.1).Balance network is balanced the assignment of terminal to concentrator.For the balancing network approximately equal number of terminals assignedto each concentrator.Penalisation is restrict the follow constraints and make solution feasible.Calculate the fitness using distance, balance and penalisation.

1.2.4 Optimal solution

A feasible solution represents the possible smallest fitness value which areminimum distance and balance the distribution of terminals among concen-trators. Feasible solution tends to the optimal solution.

1.3 Example

Available information in TA problem instance is given number of terminasand number of concentrators and locations (in the form of coordinates) ofterminals and concentrators, and individual weights of terminals and capac-ity of concentrators. The main aim of this problem is minimizing the cost.Euclidean distance equation is equation for calcuation distance between twolocations. In this problem is calculating distance between terminals to con-

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(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) Concentrator DataConcentrator Capacity Coordinates

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

Table 1.1: Example of TA problem

2 1 2 2 2 3 3 1 3 1

Table 1.2: Terminal representation with assigned concentrator

centrator which is two locations one for terminals and another for concen-trators. The cost calcuation method is calculating the euclidean distancebetween terminals to assigned concentrator[1]. Any solution for this problemmust follow the TA problem constraints which has been defined in previ-ous section. Table (1.1) indicates a given collection of terminals T=10 andgiven collection of concentrators C=3. The weight requirement of each ter-minals and coordinates of terminals are defined in Terminals Table 1.1(a).Coordinates and capacity of concentrators are defined in Table 1.1(b). Inthe figure(1.1) representation of terminals and concentrators are acording tocoordinates. Assignment of the terminals are not easy beacause more of theterminals around of the concentrator and some concentrator have not nearany terminals. According to distance are not easy to assign terminal. In Ta-ble (1.2), the solution represents the TA problem corresponds to a terminalusing integer vectors. The values i corresponds to specifies the concentratorthat terminal i is assigned to concentrator.

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Figure 1.1: A simple example of TA problem

1.4 Greedy Algorithms for TA problem

A greedy based heuristic algorithm for the TA problem is to assign terminalto nearest available concentrators in a greedy fashion. Availability refersto the ability of the concentrator to service terminal capacity requirements.For each terminal, the algorithm looks for the concentrator that is closestto the terminal. Checks if there is enough capacity to satisfy requirementsof particular terminal[14] and also satisfy the balance function or minimizethe fitness value. If there is, then the terminal is assign to this concentrator.If concentrator can not handle the terminal, the algorithm looks for thenext closest concentrator and perform evaluation[16],[17],[18]. This processrepeated until available concentrator is found and the algorithm continues toassign the remaining terminals, if there are any. otherwise, no concentratorholds the required capacity and attempt is declared a failure and solutionis infeasible. The assignments are carried out random fashion; i.e., terminalselected to be connected next is selected at random.

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Algorithm 1 Greedy ProcedureInput: terminals and concentrators coordinates, capacity, weights.Output: Minimum fitness value.

1: for all terminals do2: Calculate distance terminal to each concentrator using equation(1.1).3: Arrange concentrator according to minimum distance and balance the

assignment.4: if constraints satisfied assign terminal to lowest distance and more bal-

ance concentrator otherwise assign next lowest distance and balance.5: Calculate fitness using fitness equation.6: Every iteration update more fit value.7: end for8: return minimum fitness value.

Using the algorithm(1) solve the TA problem only minimized the Eu-clidean distance of the terminals to assigned concentrator. Sometime assign-ment of terminals to concentrators are failed due to parameter value of TAproblem are insufficient. It can fail to procedure a feasible solution when:

• The total concentrator capacity is less than total terminal capacityrequirement, or

• there is not a feasible solution to the problem instance, or

• The algorithm misses the feasible solution(s).

1.5 Greedy Anylysis

Greedy solution is based on the minimum distance of the terminal to con-centrators. Method proceeds minimum distance first then next minimumdistance for assign all terminals which not violate the constraints of the TAproblem. In this method produce results are not optimize without the helpof any heuristics.

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Chapter 2

Ant Colony Optimization

2.1 Introduction

Ant Colony Optimization (ACO) technique is used for finding the shortestpath. It has been motivated by the behavior of real ants exercises and theirforaging conduct. Ant Colony Optimization is based on the foraging behaviorof real ants. Real ants used the chemical substance for pheromone in ACOtechnique, artificial ants used a numeric value for the pheromone[13],[24],[22].One of the fundamental thoughts is the indirect communication among theants of a colony of agents, called artificial ants. It is based one sort of achemical substance, called pheromone, which used by real ants for commu-nication. The artificial pheromone are one sort of circulated numeric data(called stigmergic data) which is modified in each cycle by the ants to mir-ror their deposited and evaporation while solving a specific issue. The ACOmetaheuristic has been proposed to give an appropriate structure to largenumbers of applications of ants framework calculations for assignment ofcombinatorial optimization issues[22].

Usually, a problem to be solved by an ACO is modeled as a graph. TheACO algorithm follows an iterative process. During each iteration, a set ofartificial ants (ants for short) is distributed on this graph to perform ran-domized walks on it in search of good solutions. Individual ants, in general,do not have the capacity to find good solutions on their own. However, co-

7

CHAPTER 2. ANT COLONY OPTIMIZATION

operative interaction among ants leads to emergence of good solutions. Tworules are responsible for this cooperative interaction. The first rule called theprobabilistic state-transition rule is used when an ant has to take a decisionabout the next vertex to visit. The second rule is the pheromone-updaterule which changes the preference degree of a vertex based on the qualityof solutions contaning that vertex. Different ACO algorithms emerge fromvariations in these two rules.

2.2 ACO Metaheuristic

A standout amongst the most behavior of ants to locate the shortest pathwith the help of pheromone between their home and a food source. Fromthese studies and perceptions took after the principle algorithmic modelsfor the foraging behavior of ants. From that point forward, research in thedevelopment of ant colonies algorithms has been picked up a large numberof algorithms and applications[13][24]. An algorithm that was developed asa result of studies of ant foraging behavior is referred to as instances of theant colony optimization metaheuristic. ACO metaheuristics, with a focus onthe basic principles of ACO algorithms.

2.2.1 Behavior of Ants

It is an important issue of ants mechanism, how do ants locate the shortlength path from their nest to food source, without any help (i.e., visible,central, active coordination) ? Several investigations of the foraging behaviorof many similar kinds of species of real ants revealed starting random orlongest activity pattern in the search for finding food. As possible early findthe food source, action designs turn out to be more sorted out with moreants taking the same path for the food source. Amazingly, soon all ants takethe same path that is the shortest path.

This developing behavior of ant result providing a help for establishinga system where ants find a food source and attract other ants towards thatfood source. The establishment of systems are differing for various species. It

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Figure 2.1: Shortest Path Selection by Ants[13]

can either be in the form of direct communication or indirect communication.A large number of similar types of ants species began finding food after thatants use the pheromone value establish of paths, which is communicationvia pheromone trails[13]. In proceeding ant system, when an ant searcha food source, that ant carries a bit of food to the nest. During carries foodpiece lays chemical substance (pheromone) alongside the trail. Accordingto the value of pheromone, forager ants take decision which path is bestor shortest. Which path should follow, all paths have different pheromonevalue. Paths with a higher pheromone value which is a more probable beingselected by ants. The large number of ants follow a specific trail, on thebasis large amount of pheromone deposited by ants which path followed bymost of ants. The collective behavior of ants resulted as a behavior of ants,where positive feedback of a selected shortest path that causes of path to befollowed by most number ants. The indirect communication of ants whichmodifies their environment (laying of pheromones). In the figure(2.2) antsfind the shortest path after iteration by iteration.

2.2.2 Artificial Pheromone

Stigmergy is one type of systems that animals used for animal-to-animal com-munications. In past term stigmergy uses as a way of indirect communicationor link up to animals in their environment[13].

An example of ant activities that are linked through sematectonic stig-

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Figure 2.2: Ants finding shortest path

mergy which is including in the nest building, food sorting, and nest cleaning.Sign based stigmergy give corresponds communication through a signalingsystem which is implemented and executed on the basis of chemical substancelays by ants and deposited on each visited link. According to the exampleforaging behavior of ants inference to ants pheromone which deposited byants.

2.2.3 Simple Ant Colony Optimization

For the Simple ACO technique, each edge is assigned a small random valueto indicate the initial pheromone, τij(0). Strictly speaking, edges do not haveany pheromone concentrations for the starting or first step. An ant randomlyselects which edge to follow next. Using the simple decision logic summarizedin Algorithm, implementation is made easier by initializing the pheromoneconcentration on each link to a small random positive value.A number of ants, k = 1, . . . , nk, are placed on the source node. For eachiteration of Simple ACO, each ant incrementally and continuously constructsa path which is solution to the destination node. At every node and every

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ant executes a decision using pheromone to determine the next step or nextlink of the path. Assume that ant k is currently at node i, it selects the nextneighbor node j ∈ Nk

i , which is based on the rule of transition probabilitycalculated by the use of neighbors of node,

pkij(t) =

ταij(t)∑

j∈Nki

ταij(t), if j ∈ Nk

i

0, if j /∈ Nki

(2.1)

where Nki is the number nodes connected to node i which called neighbors

nodes of node i, with respect to ant k. If, for the node i and ant k, Nki = φ,

then predecessor or previous node to node i is included in Nki . One important

point of this if the cause loops to occur within the constructed paths restrictedthese types of loops and these loops removed once the destination node hasbeen reached. In the above equation(2.1), α is a +ve constant which is usedto amplify and influence of pheromone value concentrations.

All ants have been constructed a complete path once times from the ori-gin node to the destination node, and removed all existed loops, every antretraces constructed path reverse (from destination to the source node) de-terministically, and deposits a continuesly pheromone amount,

∆τ kij(t) ∝1

Lk(t) (2.2)

to every link of node, (i, j), of the constructed or corresponding path. Lk(t)is the length or distance of the corresponding path constructed by an ant kat time step t.

τij(t+ 1) = τij(t) +nk∑k=1

∆τ kij(t) (2.3)

where nk is the number of ants.In the equation(2.2), the aggregate pheromone esteem on a specific connec-tion is relative to the attractive quality of those ways in which the connec-tion happens in view of the most limited separation of the comparing way.The deposited pheromone, ∆τ kij, calculated using equation(2.2), evaluates

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the quality of that solution. Using Simple ACO algorithm, optimized thesolution (the corresponding constructed path) is evaluates as the reverse ofthe way length as far as the quantity of hubs in the way. Another measure-ment also can be used, i.e the length of the path, or the distance of the path.Simply, if xk(t) indicates one solution at time t, then f(xk(t)) indicates theone of the better quality of the solution. If ∆τ k not directly proportionalto the solution and all ants deposit the equal amount of pheromone then itis only the differentiate the path according to distance. The path selectionaccording to the minimum length path.

Pheromone intensities on all connections or link are permitted to evapo-rate at each iteration of the calculation before being support on the premiseof the recently picked path. For each link, (i, j),

τij ← (1− ρ)τij(t) (2.4)

with ρ ∈ [0, 1]. The constant, ρ, specifies the evoporation rate or operatorof pheromones, it is used for control the biaseness. In the alternatively, ρremove the previous decision of ants. ρ, evoporate fast of large value ofpheromone evaporation, while for smaller value evaporation rates is slower.If the pheromones evaporation is more then random search maximize andfacilitating good exploration. For ρ = 1, then search is totally random.

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2.3 ACO Algorithm

Algorithm 2 ACO AlgorithmInput: vertex, edge, cost of edge, random value of τ , number of ants, numberof iterations.Output: minimum cost path.

1: initialize τij(0).2: let t=03: ant nk on the origin node or nest.4: for all ant k=1,........,nk do5: construct a path xt(t); xt(t) = φ6: select next node on the basis of probability using equation(2.1).7: add link (i,j) to path xk(t).8: goto step 6 until destination node has been reached.9: remove loops from xk(t).

10: calculate the path length f(xk(t))11: end for12: for all link (i,j) of the graph do13: pheromone evaporation; reduce the pheromone value τij(t) using

equation(2.4).14: end for15: for all ant k=1,.......,nk do16: for all each link (i,j) of xk(t) do17: calculate ∆τ k = 1

f(xk(t))18: update τij using equation(2.3).19: end for20: end for21: t=t+122: goto step 4 until stoping condition is true.23: return the path xk(t) with smallest f(xk(t)) as the solution.

This ACO algorithm find the shortest path from source to destination inany graphs.

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Chapter 3

ACO for Terminal AssignmentProblem

3.1 Introduction

Terminal assignment problem determining minimum distance and balancethe assignment of terminals. In TA problem formed a network by connect-ing a given collection of terminals and a given collection of concentrators ofthe telecommunication network. Given terminals assigned to a given con-centrators. Terminal coordinate values and concentrators coordinate valueshave fixed locations which are known. The weight or capacity of each termi-nal and capacity of concentrator are also known that may vary from one toanother. The assignment of terminals to concentrator are the linking eachterminal to the concentrator. This problem is identified each terminal to theconcentrator which it should be assigned, under the constraints[18], in orderto minimize the fitness values.

An ant colony optimization algorithm is an edge based probabilistic tech-nique for solving computational problems which can be reduced to complexityfinding shortest paths through networks or to apply ACO methodology, theoptimization problem is transformed into the problem of finding the best pathon a weighted graph[19]. Weight defined in different problem (graph) differ-ent way, such as in terminal assignment problem distance play major role

14

CHAPTER 3. ACO FOR TERMINAL ASSIGNMENT PROBLEM

between terminals and concentrators. It is a population-based metaheuris-tic that can be used to find approximate solutions to difficult optimizationproblems[22]. Every iteration pheromone value is increase by ∆τ only on theselection edge. Which edge select more times pheromone value will be high.So, ACO also called edge based probabilistic techniques.

This algorithm, ACO is a probabilistic method and based on probabilityto solve TA problem. A computational result of ACO method compare withLSGA[8], TS[5] and ABC[11] all approaches are proposed by E.M Bernadinoet al. LSGA is also a population-based algorithm, it is extended algorithmof Genetic Algorithm (GA) to solve TA problem. LSGA is combination ofLocal Search (LS) and Genetic Algorithm (GA) for improvement of fitnessvalue of optimizing solution. Tabu Search algorithm is one of the simplestalgorithms for solving TA problem and execution time of Tabu search is alsois less compared to others. It is also proposed by E.M Bernadino et al[5].Swarm intelligence is a population-based algorithm which based on socialbehavior of animals. Bees algorithm is also based on behavior of bees andmimicking the social behavior of insects. Artificial Bee Colony (ABC) arebased on the behavior of real bees these are mainly three types Employedbees, Onlooker bees and scout bees. In algorithm implement the function ofbees with LS for optimizing solution which is bees algorithm[11].

3.2 ACO Equations

ACO equations used for solving terminal assignment problem are followings.

pij(t) =

ταij(t)η

βij(t)∑

i∈Nji

ταij(t)ηβij(t)

, if i ∈ N ji

0, if i /∈ N ji

(3.1)

N ji Number of terminals connected to jth concentrator. ηij is defined by

the terminal ith is assigned to concentrator jth and distance to terminali to

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concentratorj is distanceij calculate using the equation(1.1).

ηij = 1distanceij

In the euqation(3.1) t is defined the value is current time next time (t+1)value change or may be same. α and β are positive constatnt. τ is pheromonevalue on each terminal and concentrator edge(link).

∆τ = 1(10 + C(ibest)− C(best)) (3.2)

Where C(ibest) is the best solution of iteration and C(best) is best solutionfound since the beginning[22].

τij(t+ 1) = τij(t) + ∆τij(t) (3.3)

Pheromone value update in the above equation where i, j, t, τ representterminal, concentrator, current time value and pheromone value.

τij ← (1− ρ)τij (3.4)

In the above equation ρ value will be ρ ∈ (0 ≤ ρ < 1), we are substracting1-ρ. So, value of ρ will be minimum for sustaining the maximum iteration,it will be around of 0.013.

3.3 Basic steps

1. Pheromone InitializationInitially, we initialize the pheromone value by small random value atthe time t=0. Initialize value will be grater than 0.

2. State transitionFinding the probability using probability equation(3.1) which decidemove terminal to concentrator. Each terminal assign according to theprobability of concentrator. probability is proportional to pheromone

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concentration on concentrator node. probability depend on the distanceof the terminal and concentrator. capacity of concentrator major rolefor calculating probability.

3. Update pheromonePheromone update by the value of ∆τ which calculated by the equa-tion (3.2) pheromone is increased by 0.1 during each iteration whichterminal are selected most time pheromone value will high.

4. Reduce pheromoneReduce the pheromone value using the rho (ρ), its value lies in 0 to 1.When the program use for the huge networks then the updated value ofτ will be very high after some iteration which may be handle difficult.So, every iteration we evaporate the value of τ multiply by 1 − ρ andupdated.

5. Cost function and constraintCalculate the cost which is a summation of the Euclidean distanceof terminal and concentrator. The distance of respective terminal andconcentrator, which is only terminals distance to one connected concen-trator. Summation of terminals weight does not exceed to concentratorcapacity which to connected.

6. continuationProgram iteration until minimum cost achieved or minimum cost valueis not changed. Previous cost value is not better than current costvalue after some iteration. Many iteration cost value are not better toprevious iteration or same cost value getting then stop the iteration.

3.4 Fitness Function

Fitness function or evaluation function evaluates the solutions and plays arole in directing the search process towards the optimal solution. The givenevaluation function assesses the solution efficiently w.r.t to the other solutionsuntil the optimal solution is reached. We have made use of the exactly same

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fitness function as used in Bernadino et al. [6],[12]. This fitness function isbased on the following:

1. the total number of terminal assigned to a concentrator. The objec-tive of this equation is to balance the distribution of terminals amongconcentrators.

2. euclidean distance between assigned terminals to the concentrator. Thepurpose of this equation is to minimize the distance between the as-signed terminal and the concentrator.

3. penalization function equation is used to penalize those solutions whichare infeasible. The primary objective of this equation is to prevent thegeneration of infeasible solutions.

totalc =n∑t=1

1, if(c(t) = c)

0, otherwise(3.5)

This equation(3.5) counts the number of assigned terminals to each concen-trator. Here, ‘c’ represents a concentrator, ‘t’ represents a terminal and ‘c(t)’represents the concentrator to which terminal ‘t’ is assigned. If the terminal‘t’ is assigned to the concentrator ‘c’, add 1, otherwise add 0. This process isrepeated for each concentrator to get the total number of terminals assignedto each.

balc =

10, if(totalc = Nmax)

20 ∗ |(Nmax − totalc)|(3.6)

Above equation(3.6) is used for measuring whether the assignment of ter-minals to the concentrators is balanced or not. where Nmax is defined inequation(4.1), and ‘c’ represents the concentrator. If a concentrator is bal-anced add 10 for that concentrator and if it is not balanced then multiply by20 the absolute value of the difference between Nmax and totalc.

Penalisation =

0, if(Feasible)

500, Otherwise(3.7)

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The equation(3.7) is to penalise a solution if the assignment does not satisfythe TA problem constraints.

fitness = 0.9 ∗m∑c=1

balc + 0.1 ∗n∑t=1

distancet,c(t) + Penalisation (3.8)

The eqation(3.8) is the fitness function equation which is the summation ofequations 3.6, 1.1 and 3.7 with some multiplying factor. Here distancet,c(t)is the euclidean distance between terminal ‘t’ and the concentrator c(t) towhich this terminal is assigned.

3.5 Selection Process

Terminal Assignment Problem is assign terminals to concentrator under someconstraints and achieves optimized result also. In the whole process one isimportant problem is how to assign terminal to concentrator which is somerule defined or used in this section. Selection process putting right terminalto right concentrator.

3.5.1 Pseudo Random Proportional Rule

During the construction or make a new solution the state transition ruleis the phase. In the phase, each ant decides the assignment of terminalsto concentrator. A new state transition rule has been introduced calledpseudo-random proportional. The pseudo random proportional rule is a rulecompromise between the pseudo-random states which is choice based rule.By using the pseudo random rule the select state which is the best in terms ofprobability occurs value less than equal to q0 (exploitation) while a randomstate chosen in terms probability 1-q0 (exploration)[3].

Concentrator(j) =

maxij{ταij ∗ η

βij} if q ≤ q0

RouletteWheelProcess otherwise(3.9)

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In equation(3.9) q0 is constant value and q is randomely generated value.q,q0 ∈ [0,10] and q0 will be the middle value of the lower and upper range.

3.5.2 Roulette Wheel Process

Roulette Wheel is a selection process, it selects on the basis of probability.In roulette wheel selection process the terminals are mapped to contiguoussegments of a line. Such that each terminal segment has a range which isaccording to probability to its fitness. A random number is generated andthe match the random number to terminals range whose segment spans therandom number is selected. This process which probability value will behigh that will more probable to select and less probability value will be lessprobable to select.

Concentratorij = pijM∑l=1

pij

∗ 100 (3.10)

In the equation(3.10) i terminal, j concenterator, M is total number of con-centrator. For terminal ith, select concentrator j. pij is ACO probabilityof ith terminal and jth concentrator. In the figure(3.1) define the range of

Figure 3.1: Eaxmple of Roulette Wheel Process

concentrator according to the value of equation(3.10). Firstly perform theoperation on all concentrator for the terminal ACO using equation(3.10),sequentially perform addition operation for all concentrator and mark uplower and upper range for the each concentrator. Upper range of the lastconcentrator also the range of the value means no upper value generate therandom value. Every time generate random value and decide which concen-trator value range belongs to its value select that concentrator. Figure(3.2)

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Figure 3.2: Concentrator selection probability

shows chances(percentage) of the concentrator selection. Higher range orpercentage value is more probable to select concentrator.

3.6 Boosting Process

The main aim is boosting algorithm finding the promising states that maybe close to the solution[23]. Many times problem stuck on the local minimawhich tend to global minima. For solving Terminal Assignment Problemrequires boosting for getting global minima, in this problem are more localminima and solution is stuck in many local minima. For solving these things,we require four components [ n, m, u, v ].where,

• n is a number of terminals.

• m is a number of concentrators.

• u has measured the assignments.

• v is the tightness of the assignments.

Assignments of terminals to the concentrator processed any algorithm andmeasure the tightness using the equation according to the problem. For

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boosting used Local Search algorithms and get initial states of the solution.In Terminal Assignment Problem using the probability calculating for thetightness terminal to the concentrator and then apply proposed algorithm.

3.7 Algorithm

Algorithm 3 Solve TA problem using ACOInput: terminal and concentrator data, number of iteration, number of ants,α,β.Output: fitness value.

1: Read data.2: Calculate Euclidean distance each terminal to each concentrator.3: Initialize pheromone value.4: Boosting using local search and store fitness value.5: for all iteration do6: for all ants do7: Calculate ACO probability.8: Pheromone evoporate.9: Pheromone update.

10: Select concentrator for each terminal by using selection process.11: Calculate fitness.12: Update better fitness.13: end for14: end for15: return fitness value.

3.8 Computational Result

This experiments performed on nine standard benchmark TA instances, col-lection of TA instances of different size. We have taken instaces from litrature[9].To compare our result to the result of Discrete Differential Evaluation Algorithm[9]proposed by Bernadino et al. because they

• Used same set of instances.

• Used same evaluation or fitness function.

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Table 3.1: Results - Compare Bets Fitnesses and Execution Time

Instance LSGA TS ABC ACOBestF 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 278.65 <1s4 286.89 <1s 286.89 <1s 286.89 <1s 297.23 <1s5 335.09 <1s 335.09 <1s 335.09 <1s 338.79 1s6 371.12 1s 371.12 <1s 371.12 <1s 386.40 1s7 401.21 1s 401.49 1s 401.21 <1s 417.09 3s8 563.19 7s 563.34 1s 563.19 3s 581.46 7s9 642.83 7s 642.86 2s 642.83 4s 662.38 7s

• Used same programming language for implementation.

• The same way representation of terminals and concentrators.

Terminal Assignment Problem is solve using the ACO technique and resultfor all instances are given below table(3.1). In result table compare the resultwith LSGA[8], TS[5] and ABC bees algorithm[11] algorithm result. Theseare all approaches proposed by the Bernadino at al.

Computational result performed on nine standard instances which havetaken from the comparing approach literature. For the implementation ofprogram used same environment. The proposed algorithm have been ex-ecuted utilizing a processor Intel Core Duo T2300. The Ts (Run Time)compares to the execution time that every algorithm needs to acquire thebest practical arrangement. Our methodology has been implemented in Cand executed on an Intel Core 2 Duo (T2300) framework with 2 GB RAMrunning at 3.0 GHz under Ubuntu working framework. Initial value of ACOmethod are number of population (ants) 10, number of iteration 100, ρ value0.01, α value 1 and β value 2. Table (3.1) present Best Fitness (BestF) andTime in seconds (Ts) seprately for all methods LSGA, Tabu Search, Beesalgorithm based on ABC and ACO. Table (3.2) present average fitness value(AvgF) and standard deviations (std) seprately for all methods LSGA, TabuSearch, Bees algorithm based on ABC and ACO.

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Table 3.2: Comparison of Average Fitnesses and Standard deviations

Instance LSGA TS ABC ACOAvgF 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 0.003 270.69 0.23 270.76 0.30 270.29 0.04 281.09 1.434 286.99 0.13 287.93 0.75 286.89 0.00 298.90 1.665 335.99 0.60 335.99 0.59 335.11 0.02 338.79 0.006 371.68 0.24 372.44 0.45 371.21 0.10 389.29 1.597 402.41 0.50 403.25 0.73 401.45 0.12 421.13 2.008 564.94 0.52 564.50 0.54 563.37 0.11 581.46 0.009 646.52 0.84 644.18 0.48 643.41 0.25 662.38 0.00

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Chapter 4

Modified Vogel ApproximationMethod

4.1 Introduction

For the latest development of telecommunication, their expanding many-sided quality and the heterogeneity of the connected component, and com-binatorial optimization issues showed up. Terminal Assignment is a knownproblem in telecom systems. For telecommunication problem plays majorrole of execution time. The main task here is to assign a given arrange-ment of terminals to a given arrangement of concentrators. The objectivesproblem optimally arrangements of assignment and minimize the distanceamongst concentrators and terminals assigned to them.

Vogel Approximation Method (VAM) is a heuristic and more often thannot gives a superior beginning arrangement than other strategies. Use ofVAM to a given issue does not ensure that an ideal arrangement will comeabout. Notwithstanding, a great arrangement is constantly acquired withrelatively little exertion. Truth be told, VAM used for yields initial solutionwhich is nearer to optimal solution. It is also estimated for transportation,arrangement, assignment problem.

This algorithm is based VAM which is modification of VAM according tothe TA problem environment. In our implementation modify VAM to solve

25

CHAPTER 4. MODIFIED VOGEL APPROXIMATION METHOD

TA problem and compare to HDE[7],MHDE[10] and DDE[9] all algorithmhave been proposed by E.M Bernadino et al. HDE utilizes a Differential Evo-lution Algorithm (DE) to investigate a few locales of the hunt space and at thesame time consolidates a system (LS) to heighten the pursuit around somechose districts. MHDE is based on Hybrid Differential Evaluation methodwhich is a new multiple strategies with improvement of LS procedure. DDEalgorithm is improvement version of DE algorithm. It is also a combinationof DE and LS with some recombination of steps with new trial individualsmutated for target individuals.

4.2 Vogel Approximation Method to SolveTerminal Assignment Problem

For solving the TA problem, the concept has extracted from the Vogel Ap-proximation Method (VAM) and incorporated in the algorithm. It is anefficient method for the initial solution and is closer to the optimal solution.Using this approach, the terminals are assigned to the concentrators whichsatisfy the TA problem constraints and finds the initial solution[15].

Nmax = round( nm

) + 1 (4.1)

Where, Nmax is the maximum number of terminals assigned to one concen-trator. n is the total number of terminals and m is the total number ofconcentrators.

The method proceeds by calculating the arithmetic difference of the twoleast distance concentrators for each terminal. Then it selects the terminalwith the maximum difference assigns concentrator having least distance fromit. It then selects the next unassigned terminal with a maximum differenceand assigns it to the concentrator having least distance from the correspond-ing terminal. This process repeats until assigning Nmax terminals to anyconcentrator or until the concentrator capacity is less than all unassignedterminals weights. Every assignment subtracts the corresponding terminal

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weight from the assigned concentrator. Mark the concentrator which is al-ready assigned for Nmax terminals or no remaining capacity is greater thanor equal to any unassigned terminal. Repeat the process until all terminalsare assigned to the concentrator, the marked concentrator is exempt fromthe remaining process.

Algorithm 4 Vogel Approximation Method concept to Solve TA problemInput: terminal and concentrator data, Nmax.Output: fitness, assigned data.

1: Read a given instance.2: Calculate distance of each terminal from each concentrator using

equation(1.1).3: Calculate the difference between two lowest distance concentrators.4: Select maximum difference terminal.5: Assign selected terminal to the lowest distance concentrator.6: Subtract terminal weight from the assigned concentrator for every as-

signment process.7: Select next maximum difference terminal and assign it to the lowest dis-

tance concentrator. This process repeats until Nmax terminals are as-signed to any concentrator, or the concentrator capacity is exhausted.

8: Mark concentrator assigned Nmax terminals or no more capacity remains.9: Marked concentrators are exempt from continuing process.

10: Repeat the steps 3 to 9 until all terminals are assigned.11: Calculate fitness.12: Return fitness and assigned terminals data.

4.2.1 Selection Procedure

The roulette wheel procedure is a likelihood based choice technique. In thismethodology, the concentrator whose likelihood high is more plausible to beselect for a terminal. Every terminal produces an arbitrary number which co-ordinated with the scope of concentrators. Return that concentrator matches

page 27


with the randomly produced number.

concentratortj = 1m∑j=1

distancetj

∗ 100 (4.2)

Where, distancetj is euclidean distance between terminal t to concentra-tor j which is defined in equation(1.1), concentratortj is concentrator j forterminalt. Multiplying by 100 gives converting the probability (fractional)value in integer form.

In case, the selected concentrator does not satisfy the constraint of TAproblem then directly select highest value of 1

distancetjwhich satisfy the TA

problem constraints.

4.2.2 Evaluation Function

Evaluation function or assessment capacity assesses the arrangements andmoves towards the ideal arrangement. The given assessment capacity eval-uates the arrangement effectiveness of w.r.t to alternate arrangements untilthe ideal arrangement came to. This assessment capacity is depicted inthe writing of E.M. Bernadino et al. in all the proposed method literaturewhich are LSGA[8], GAMO[4], TS[5], ABC[11], HDE[7], MHDE[10], DDE[9],HACO[6], NHACO[12]. This fitness function is based on the following:

1. the aggregate number of terminal allocated to a concentrator. The goalof this condition is to adjust the appropriation of terminals amongconcentrators.

2. Euclidean separation between allocated terminals to the concentrator.The motivation behind this condition is to minimize the separationbetween the doled out terminal and the concentrator.

3. Punishment capacity condition is utilized to make the arrangement at-tainable. The essential goal of this condition is to quit surpassing thetotal weight of doled out terminals to the concentrator limit.

Fitness function and its equations have describe in section(3.4).

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4.2.3 MVAM Algorithm

This approach, Modified Vogel Approximation Method (MVAM) algorithmhas embedded with these methods: Vogel Approximation Method concept,and Local Search. Vogel Approximation Method concept is using for thestarting solution and then its output is passed to the next process. LocalSearch used in each iteration for optimizing the result.

Algorithm 5 MVAM algorithmInput: number of iteration, Nmax,p,vloop.Output: fitness.

1: Apply algorithm(4).2: Forward fitness and assigned terminals.3: for all iterations do4: initialize p to zero.5: while p < vloop do6: Apply Local Search.7: Select concentrator for each terminal by using selection process.8: increment value of p by one.9: Calculate fitness.

10: end while11: Compare fitness.12: Update better fitness.13: end for14: return fitness.


This trials performed on nine TA occurrences, accumulation of TA examplesof various size. We have taken instances from literature. To contrast our out-come with the consequence HDE[7], MHDE[10], DDE[9] algorithm proposedby Bernadino et al. since they

→ Work on same instances.

→ Apply same fitness function.

→ Implementation and execution in same programming language.

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Table 4.1: Best Fitnesses and Time Execution compare to different ap-proaches

Instance HDE MHDE DDE MVAMBestF 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 <5s 270.26 <1s 270.26 <1s 271.07 <1s4 286.89 <5s 286.89 <1s 286.89 <1s 288.07 <1s5 335.09 <5s 335.09 <1s 335.09 <1s 335.88 <1s6 371.12 58s 371.12 <1s 371.12 <1s 372.34 <1s7 401.21 118s 401.21 2s 401.21 <1s 404.25 <1s8 563.19 274s 563.19 10s 563.19 2s 563.83 2s9 642.83 456s 642.83 15s 642.83 3s 644.48 2s

→ Same format used for representing terminals and concentrators.

Computational result performed on nine standard benchmark instances whichhave taken from the contrasting methodology writing. For the execution ofsystem utilized same environment. The proposed method have been exe-cuted using a processor Intel Core Duo T2300. The Ts (Run Time) thinksabout to the execution time that each calculation needs to gain the best vi-able game plan. Our procedure has been actualized in C and executed on anIntel Core 2 Duo (T2300) system with 2 GB RAM running at 3.0 GHz underUbuntu OS. For execution of this method some initial value which is numberof iteration 1000 and vloop value 4. Table (4.1) present the Best Fitnessvalue (BestF) and Time in second (Ts) seprately for all methods Hybrid DE,Multiple strategy of HDE, Differential DE and Modified VAM.

Table (4.2) present average fitness value (AvgF) and standard deviations(std) separately for all methods Hybrid DE, Multiple strategy of HDE, Differ-ential DE and Modified VAM. We are getting the result w.r.t average fitnessis around of same to previous approaches HDE, MHDE and DDE. Standarddeviations of our approaches are zero for all the instances otherwise, the otherapproaches standard deviations are still a fraction value for some instances.In terms of time in seconds (Ts) is less compared to all approaches means

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Table 4.2: Average Fitnesses and Standard deviations compare to differentapproaches

Instance HDE MHDE DDE MVAMAvgF 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 0.003 270.35 0.06 270.75 0.15 270.47 0.22 271.07 0.004 286.97 0.09 287.17 0.14 286.89 0.00 288.07 0.005 335.42 0.16 336.55 0.39 335.26 0.17 335.88 0.006 371.60 0.17 373.19 0.42 371.38 0.22 372.34 0.007 401.58 0.12 403.61 0.33 401.62 0.28 404.25 0.008 564.03 0.21 572.04 0.76 564.07 0.38 563.83 0.009 646.65 0.61 648.46 0.48 643.96 0.46 644.48 0.00

our approach MVAM is taking less time for execution.

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Chapter 5

A Hybrid Ant ColonyOptimization Approach

5.1 Introduction

In the present times, telecommunication is the heart and soul of the way inwhich people connect with each other. Terminal Assignment problem is anNP-hard combinatorial optimization problem[2]. The objective of the TAproblem is to minimize the weighted sum of distance of terminals from theirassigned concentrators and the unbalance in loads on different concentratorsmeasured by a balance function.

We have developed a hybrid ant colony approach for the TA problem. Ourapproach is a combination of Vogel approximation method, an ant colonyoptimization algorithm and a local search. Vogel Approximation Method(VAM) has used in our approach for the initial solution[15] as the solutionsobtained by this method is near to the optimal solution in most cases. Wehave applied a local search on solutions obtained through ACO. Applicationof Local Search (LS) helps in further improving the solutions.

In this chapter, we have used the same objective function as usend inBernadino et al[12],[6]. Bernadino et al. first proposed a Hybrid Ant ColonyOptimization (HACO) which used ACO and Local Search[6]. Later, theyproposed another ACO based hybrid approach called New Hybrid Ant Colony

32

CHAPTER 5. A HYBRID ANT COLONY OPTIMIZATION APPROACH

Optimization (NHACO)[12]. Two variants of NHACO were presented, viz.NHACO with Local Search (NHACO-LS) and NHACO with Tabu Search(NHACO-TS)[12].

5.2 Important ACO Equations

Ant Colony Optimization is an efficient technique of Swarm Intelligence,which is mostly used for finding the shortest path. It is a population-based metaheuristics technique which used for finding the optimal solu-tion of computational intelligence problems. This technique is based onthe real ants behavior. The ants find the shortest path from the nest totheir food sources after some iterations. Real ants find the shortest pathwith the help of pheromone which is a chemical substance. For the artificialants used a numeric value which works as the pheromone and changes periteration[13],[24],[22] according to the solution found during that iteration.

Some ACO equations are used in our proposed algorithm for solving theTA problem. The first equation is the ACO probability equation.

Pij = τijn∑i=1

eji ∗ τij(5.1)

where, Pij is ACO probability, τij is pheromone value for assignment ofterminali to concentratorj, n is total number of terminals, eji is a binarycoefficient[20]. If terminali is assigned to concentratorj in the solution gen-erated in the previous iteration (in the very first iteration, it is the solutiongenerated by Vogel approximation method), then eji is 1 otherwise 0.

Vij = totaltj ∗ Pij (5.2)

Where, Vij is the probability multiplied by trade-off parameter totaltj onterminali to concentratorj, totaltj is the total number of terminal assignedto concentratorj.

Pheromone evaporation equation is used to simulate the evaporation of

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pheromones in real ants. Evaporation of artificial pheromone values is re-quired to forget the bad choices made previously. The evaporation rate ofpheromone is same for every iteration.

τij = (1− ρ) ∗ τij (5.3)

Where, ρ is the evaporation rate or evaporation operator.Pheromone value is updated by a small amount (∆τ) per iteration for chang-ing the preferences of terminals for various concentrators as per the solutiongenerated in the previous iteration.

τij = τij + eji ∗ k ∗1

distanceij(5.4)

where k is a positive integer, distanceij is the distance between terminali

and concentratorj. Please note that only pheromone values correspondingto assignments present in the solution will be augmented as eji = 1 for theseassignments only. It is imperative to mention here that in our approach eachant updates the pheromone instead of commonly used policy where only bestant updates the pheromone.

5.2.1 Local Search

Local search is used for improving a solution according to the given problem.It is applied in every iteration. The purpose of applying Local Search (LS) inthis problem is to minimize the distance and balance the number of terminalsassigned to various concentrators. We have used two local searches which areused one after the other. The first local search is derived from exchange basedlocal search used in [6],[12].

The first local search involves exchanging assignment of terminals be-tween two distinct concentrators. In this process, the distance between theterminals and corresponding assigned and unassigned concentrators are cal-culated respectively. The assignment of two terminal is swapped accordingto the minimum distance if this re-assignment satisfies all the TA problemconstraints.

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The second local search is used for balancing assignment of terminalsto concentrators. In this method, we compare the distance of terminalsfrom its assigned and the unassigned concentrators respectively. Then, wecompare the number of assignments on particular concentrators. If both thedistance and the number of assignments are less on unassigned concentratorin comparison to currently assigned concentrator or distance is same butnumber of assignments is less then re-assign the terminal to this concentratorprovided it satisfies all the problem constraints.

Both the local searches stop when the first improving move is found. Wehave iterated this sequential combination of these two local searches thrice.

5.2.2 Selection Procedure

The roulette wheel process is a probability-based selection procedure. Inthis procedure, the concentrator having higher probability is more likely tobe selected for a terminal. For a particular concentrator, the probability ofits selection is calculated by using equation(5.5). Based on these probabilityvalues, a distinct range is assigned to each concentrator. For assigning eachterminal, a random number is generated which is compared with the range ofdifferent concentrators. The terminal is assigned to the concentrator underwhose range the generated random number falls.

concentratortj = Vtjm∑j=1

Vtj

(5.5)

Where, Vtj is defined in equation(5.2), concentratortj is the probability thatterminalt is assigned to concentratorj.

In case, the selected concentrator does not satisfy the constraints of theTA problem then directly select the concentrator having the highest value ofVij which satisfy all the TA problem constraints.

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5.2.3 Hybrid Ant colony Algorithm

Our proposed Hybrid Ant Colony Algorithm (HACA) has been embeddedwith three methods: Vogel Approximation Method, Ant Colony Optimiza-tion, and Local Search Procedure. Vogel Approximation Method concept isused for generating a good initial solution which is used in the initial itera-tion of the ACO to compute the probability values. Local search procedure,which is composed of two local searches, is used in each iteration to improvethe solution generated through ACO. Algorithm 6 provides the pseudo-codeof HACA.

Algorithm 6 Hybrid Ant colony AlgorithmInput: number of iteration, number of ants, Nmax,τ ,ρ,k.Output: fitness.

1: Apply algorithm(4).2: Forward fitness and assigned terminals to ACO process.3: Initialize pheromone value.4: for all iterations do5: for all ants do6: Calculate ACO probability.7: Calculate value of Vij.8: Pheromone evoporate.9: Pheromone update.

10: Select concentrator for each terminal by using selection process.11: Apply Local Search.12: Calculate fitness.13: end for14: Compare fitness.15: Update better fitness.16: end for17: return fitness.


We have compared the result of our proposed algorithm with the algorithmsproposed by Bernadino et al. [6],[12]. The performance of algorithm proposed

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Table 5.1: 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

Table 5.2: Average solution quality and standard deviation of various ap-proaches

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

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in [6],[12] were tested on nine instances. The experimental results of ourproposed algorithm are compared with algorithm HACO, NHACO-LS, andNHACO-TS on the same nine instances. In these nine instances, the numberof terminals vary from 10 to 100 and number of concentrators vary from 3 to30. Like the approaches in [6],[12], our approach is executed 100 independenttimes on each instance.

For all the computational experiments, our proposed algorithm uses fol-lowing parameter values: number of ants are 10, number of iterations are200, initial pheromone value (τ) is 100, evaporation rate (ρ) is 0.01, and ‘k’constant value is 2. These values are chosen empirically.

Table 5.1 presents the result of the previously proposed algorithms HACO,NHACO-LS, NHACO-TS, and our proposed algorithm HACA in terms ofbest fitness value and execution time in seconds. The first column repre-sents the instance number, and the remaining columns represent differentalgorithms results. Each result column has two columns BestF (Best Fitnessvalue) and Ts (Run Time in seconds). It is clear from this table that allalgorithms found the same best value over 100 runs on all the instances

Table 5.2 compares the average fitnesses (AvgF) and standard deviations(Std) obtained by various approaches over 100 independent runs. The firstcolumn represents the instance number and the subsequent columns rep-resent the AvgF and Std for various algorithms. The better algorithm isdetermined by minimum average fitness value and minimum standard devia-tion. Our proposed algorithm obtained better average fitnesses, and standarddeviations in comparison to the existing algorithms particularly on larger in-stances.

5.3.1 Comparison to the previous approaches

We have compared our proposed algorithm with the existing algorithmswhich are based on Ant Colony Optimization technique. To achieve effi-ciency, local search has been used in both the algorithms. Our algorithmis different from the existing algorithm in the way that our approach usesVAM concept for the initial solution while the existing approaches use a

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greedy method or random solution. In the compared approach, the selec-tion of a concentrator for a terminal is carried out by finding a concentratorwhose pheromone value is highest for that concentrator. We have calculatedthe trade-off value of a concentrator corresponding to all terminals whichassigned to the same concentrator.

5.3.2 Result Analysis

We have designed and implemented three approaches for solving TA prob-lem. An Ant Colony Optimization (ACO), Modified Vogel ApproximationMethod (MVAM), and a Hybrid Ant Colony Algorithm (HACA) and com-pare the our result to the previously implemented approaches for TA prob-lem which are Local Search Genetic Algorithm (LSGA), Genetic Algorithmwith Multiple Operators (GAMO), Tabu Search (TS), Bee algorithm basedArtificial Bee Colony (ABC), Hybrid Differential Evoluation (HDE), Multi-ple starategy Hybrid Differential Evouation (MHDE), Discrete DifferentialEvoluation Algorithm (DDE), Hybrid Ant Colony Optimization Algorithm(HACO), New Hybrid Ant Colony Optimization (NHACO) which have twopart one is NHACO with Tabu Search (NHACO-TS) and another is NHACOwith Local Search (NHACO-LS). NHACO-TS approach is best-resulting ap-proach in all previously approaches. Compare our ACO approach to the bestresulted previous approach, in terms of all parameter best fitness (BestF),Time in seconds (Ts), Average fitness (AvgF) and standard deviations (std)are not performing better result. MVAM approach performance are betterin terms are time (Ts) and standard deviation (std) and in terms of best fit-nesses (BestF) and average fitnesses (AvgF) is slightly difference to the bestapproach. HACA approach performance is best and effectiveness on all pa-rameter BestF, Ts, AvgF and Std to all previously implemented approaches.

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Chapter 6

Conclusion and Future Work

We have designed and implemented a Hybrid Ant Colony Algorithm (HACA)for the terminal assignment problem which uses the concept of Vogel Approx-imation Method (VAM) to generate an initial solution and a local search tofurther improve the solutions obtained through ACO algorithm. To ourknowledge, VAM concept has been used for the first time to solve the termi-nal assignment problem. We have compared our hybrid approach with threestate-of-the-art approaches on standard benchmark instances of the problem.Computational results show the effectiveness of our approach.

As a future work, we intend to incorporate an appropriate heuristic termin our state transition rule to further improve the performance of our hybridapproach. Approaches similar to our hybrid approach can be developed forother NP-hard assignment problems. The manner in which we have usedVogel Approximation Method can be used for other assignment problemsalso to generate a good initial solution for them. Like Vogel ApproximationMethod, other methods available in the operations research literature forsolving transportation problem/assignment problem may be tried for TAand other related problems.

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