Abstract: Meta heuristic() 2 while (termination criterion not satisfied)


This paper will present an overview of ant
algorithms that is algorithms for colony optimization that took insight which
was observed from ant colonies .We are going to discuss about the findings on
real ants and Ant Colony Optimization (ACO) algorithm is applied to wide range
of problems like Travelling Salesman , Job-Shop Scheduling ,Vehicle Routing.

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Ant Colony Optimization is a probabilistic technique which is used for
searching optimal path in the graph based on behaviour of ants seeking a path
between their colony and source of food. It is a Meta-heuristic optimization.
It was first proposed by Marco Dorigo in 1992 as a multi-agent approach to
difficult combinatorial optimization problems such as Travelling salesman
problem (TSP), Quadratic assignment problem (QAP).

The ACO metaheuristic consists of group of
artificial ants with the characteristics to search good solutions to discrete
optimization problem. An Ant algorithm was inspired by the observation of real
ant colonies, Ants are social insects (i.e.) insects that lives in colonies and
whose behavior is directed more to the survival of the colony as a whole than
to that of a single individual component of the colony. An important and
interesting behavior of ant colonies is that their behavior and in particular,
how ants can find the shortest paths between food sources and their nest.

While walking from food sources to the
nest and vice versa, ants deposit on the ground a substance called pheromone,
forming in this way a pheromone trail. Ants can smell these this chemical and
when choosing their way tend to choose, in probability paths marked by them and
allows them to travel from food source to their colonies. Other ants from the
colony follow their nestmates to follow their paths to the food source. This is
how shortest path is emerged for their food hunting.

?rst ants to arrive at the food source are those that took the two shortest
branches, so that, when these ants start their return trip, more pheromone is
present on the short branch than on the long branch, stimulating successive
ants to choose the short branch. 

Optimization Algorithms:

Ant Colony Optimization Algorithm:

1 procedure ACO Meta heuristic()

2 while (termination criterion not

3 schedule activities

4 ants generation and activity();

5 pheromone evaporation();

6 daemon actions(); foptionalg

7 end schedule activities

end while

9 end procedure

10 procedure ants generation and

11 while (available resources)

12 schedule the creation of a new ant();

13 new active ant();

14 end while

15 end procedure

16 procedure new active ant() fant

17 initialize ant();

18 M = update ant memory();

19 while (current state 6= target state)

20 A = read local ant-routing table();

21 P = compute transition probabilities(A;
M; problem constraints);

22 next state = apply ant decision
policy(P; problem constraints);

23 move to next state(next state);

24 if (online step-by-step pheromone

25 deposit pheromone on the visited arc();

26 update ant-routing table();

27 end if

28 M = update internal state();

29 end while

30 if (online delayed pheromone update)

31 evaluate solution();

32 deposit pheromone on all visited

update ant-routing table();

34 end if

35 die();

36 end procedure



First ACO
algorithm to be proposed (1992) Pheromone values are updated by all the ants
that have completed the tour.

                                        ?ij ?
(1 ? ?) · ?ij + Pm k=1 ?? k ij

 where ? is the evaporation rate m is the
number of ants ?? k ij is pheromone quantity laid on edge (i, j) by the k th
ant ?? k i,j = ( 1/Lk if ant k travels on edge i, j 0 otherwise where Lk is the
tour length of the k th ant.



Ant System Algorithm:

                              1: for each colony do

for each ant do

                3: generate route  

                             4: evaluate route

evaporate pheromone in trails

deposit pheromone on trails

                             7: end for

                             8: end for

Ant Colony System(ACS):

First major
improvement over Ant System Differences with Ant System:

1 Decision
Rule – Pseudorandom proportional rule

2 Local
Pheromone Update

3 Best only
offline Pheromone Update

Ants in ACS
use the pseudorandom proportional rule Probability for an ant to move from city
i to city j depends on a random variable q uniformly distributed over 0, 1,
and a parameter q0. If q ? q0, then, among the feasible components, the
component that maximizes the product ?il? ? il is chosen, otherwise the same
equation as in Ant System is used. This rule favours exploitation of pheromone

component against exploitation: local pheromone update. The local pheromone
update is performed by all ants after each step. Each ant applies it only to
the last edge traversed: ?ij = (1 ? ?) · ?ij + ? · ?0 where ? ? (0, 1 is the
pheromone decay coefficient ?0 is the initial value of the pheromone

Best only
offline pheromone update after construction Offline pheromone update equation
?ij ? (1 ? ?) · ?ij + ? · ?? best ij where ? best ij = ( 1/Lbest if best ant k
travels on edge i, j 0 otherwise Lbest can be set to the length of the best
tour found in the current iteration or the best solution found since the start
of the algorithm.

Ant Colony System Algorithm:

for each colony do

for each ant do

generate route

evaluate route

5: evaporate pheromone in all trails (? rate)

deposit pheromone on all trails

end for

8: evaporate pheromone in best global route (?2 rate)

deposit pheromone on best global route

end for

Max-Min Ant System (MMAS):

by St¨utzle and Hoos (1996), as another variation for the TSP, the MMAS
algorithm shows di?erences in the steps of pheromone deposition and
evaporation, that occur only after the i-th ant for each colony stablish its

with Ant System: 1 Best only offline Pheromone Update 2 Min and Max values of
the pheromone are explicitly limited ?ij is constrained between ?min and ?max
(explicitly set by algorithm designer). After pheromone update, ?ij is set to
?max if ?ij > ?max and to ?min if ?ij < ?min Max-Min Ant System Algoritm:                          1: for each colony do                          2: for each ant do                          3: generate route                          4: evaluate route                          5: end for                         6: verify for global or local best                        7: evaporate pheromone in all trails                        8: deposit pheromone on best global route                         9: end for Application of ACO u  Traveling Salesman u  Job-Shop Scheduling u  Vehicle Routing u  Graph Coloring   Travelling Salesman Problem: The Travelling Salesman Problem describes a salesman who must travel between N cities. The order in which he does so is something he does not care about, as long as he visits each once during his trip, and finishes where he was at first. Each city is connected to other close by cities, or nodes, by airplanes, or by road or railway. Each of those links between the cities has one or more weights (or the cost) attached. The cost describes how "difficult" it is to traverse this edge on the graph, and may be given, for example, by the cost of an airplane ticket or train ticket, or perhaps by the length of the edge, or time required to complete the traversal. The salesman wants to keep both the travel costs, as well as the distance he travels as low as possible. The Traveling Salesman Problem is typical of a large class of "hard" optimization problems that have intrigued mathematicians and computer scientists for years. Most important, it has applications in science and engineering. For example, in the manufacture of a circuit board, it is important to determine the best order in which a laser will drill thousands of holes. An efficient solution to this problem reduces production costs for the manufacturer. Difficultychange | change source The travelling salesman problem is regarded as difficult to solve. If there is a way to break this problem into smaller component problems, the components will be at least as complex as the original one. This is what computer scientists call NP-hard problems. Many people have studied this problem. The easiest (and most expensive solution) is to simply try all possibilities. The problem with this is that for N cities you have (N-1) factorial possibilities. Job-Shop Scheduling Problem: In job-shop scheduling a finite set of n jobs is taken and each job consists of a chain of operations. It also consists of a finite set of m machines, each machine can handle at most one operation at a time. Each operation needs to be processed during an uninterrupted period of a given length on a given machine. Purpose is to find a schedule, that is, an allocation of the operations to time intervals to machines, that has minimal length. The job-shop scheduling problem is the problem of assigning operations to machines and time intervals so that the maximum of the completion times of all operations is minimized and no two jobs are processed at the same time on the same machine. The basic algorithm they applied was exactly the same as Ant System(AS). Job shop scheduling or the job-shop problem (JSP) is an optimization problem in computer science and operations research in which ideal jobs are assigned to resources at particular times. The most basic version is as follows: We are given n jobs J1, J2, ..., Jn of varying processing times, which need to be scheduled on m machines with varying processing power, while trying to minimize the makespan. The makespan is the total length of the schedule (that is, when all the jobs have finished processing). In most practical settings, the problem is presented as an online problem (dynamic scheduling), that is, the decision of scheduling a job can only be made online, when the job is presented to the algorithm. This problem is one of the best known combinatorial optimization problems, and was the first problem for which competitive analysis was presented. Applying machine learning to job scheduling is an emerging approach. In this approach, artificial intelligence determines optimizations without the need for human programmers to create an algorithm for them or to fully understand the complex causation that drives them. The name originally came from the scheduling of jobs in a job shop, but the theme has wide applications beyond that type of instance   Vehicle routing Problem The Vehicle Routing Problem (VRP) dates back to the end of the fifties of the last century when Dantzig and Ramser set the mathematical programming formulation and algorithmic approach to solve the problem of delivering gasoline to service stations. Since then the interest in VRP evolved from a small group of mathematicians to the broad range of researchers and practitioners, from different disciplines, involved in this field today. The VRP definition states that m vehicles initially located at a depot are to deliver discrete quantities of goods to n customers. Determining the optimal route used by a group of vehicles when serving a group of users represents a VRP problem. The objective is to minimize the overall transportation cost. The solution of the classical VRP problem is a set of routes which all begin and end in the depot, and which satisfies the constraint that all the customers are served only once. The transportation cost can be improved by reducing the total travelled distance and by reducing the number of the required vehicles. The majority of the real world problems are often much more complex than the classical VRP. Therefore in practice, the classical VRP problem is augmented by constraints, such as vehicle capacity or time interval in which each customer has to be served, revealing the Capacitated Vehicle Routing Problem (CVRP) and the Vehicle Routing Problem with Time Windows (VRPTW), respectively. In the last fifty years many real-world problems have required extended formulation that resulted in the multiple depot VRP, periodic VRP, split delivery VRP, stochastic VRP, VRP with backhauls, VRP with pickup and delivering and many others.   The vehicle routing problem lies at the heart of distribution management. It is faced each day by thousands of companies and organizations engaged in the delivery and collection of goods or people. Because conditions vary from one setting to the next, the objectives and constraints encountered in practice are highly variable. Most algorithmic research and software development in this area focus on a limited number of prototype problems. By building enough flexibility in optimization systems one can adapt these to various practical contexts.         Conclusion: In this article we have discussed about the ant colony optimization (ACO) metaheuristic and we have given an overview of ACO algorithms. The ant colony algorithms are explained and they are implemented in different problems like travelling salesman, job-shop scheduling and vehicle routing.AS is very common in the practical usage of these heuristics, ACO algorithms often end up at some distance from their inspiring natural metaphor. ACO algorithms so enriched are very competitive and in some applications they have reached world-class performance. For example, on structured quadratic assignment problems AS-QAP, HAS-QAP, and MMAS-QAP are currently the best available heuristics. In conclusion, we hope this paper has achieved its goal by implementing the algorithms like ant algorithm, ant system algorithm, min-max ant system algorithm in the different problem like travelling salesman, job-shop scheduling, vehicle routing.