International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 5, May 2012)309
A Survey Paper on Solving Travelling Salesman
problem Using Bee colony optimization
1
Anshul Singh,
2Devesh narayan
1Faculty of Computer Science and Engineering, Rungta College of Engineering and Technology, Bhilai, India 2
Faculty of Computer Science and Engineering,Rungta College of Engineering and Technology, Bhilai, India 1[email protected]
Abstract— Traveling Salesman Problem (TSP) is about finding a Hamiltonian path with minimum cost. Travelling salesman problem (TSP) finds its application in many real world industrial applications including the areas such as logistics, transportation, and semiconductor industries. few potential applications of TSP includes finding an optimized scan chains route in integrated chip testing, parcels collection and sending in logistics companies, and transportation routing problem. This paper gives a brief overview of the various approaches that have been implemented to solve TSP and specifically bee colony optimization (BCO) for solving TSP. The BCO model is constructed algorithmically based on the collective intelligence shown in bee foraging behavior.
Keywords- bee colony optimization, foraging behavior, heuristic search, travelling salesman problem, waggle dance
I. INTRODUCTION
Animals such as ant, bee, fish, and cockroach show many interesting collective behaviors. A colony of honey bees can extend itself over long distances and in multiple directions simultaneously to exploit a large number of food sources. An example will be the highly coordinated pattern shown in their foraging and aggregation behaviors. The individuals in the beehive usually perform their actions locally with limited knowledge about the entire system. However, when their local actions are combined, they emerge to produce global effects.
This paper describes the work in applying the Bee Colony Optimization (BCO) model, which is based on foraging behavior in bee colony, to Traveling Salesman Problem (TSP). This research is inspired by [1], on using a new honey bee algorithm for dynamic allocation of Internet servers. The paper starts with a discussion on TSP in Section II. Literature survey has been shown in section II. A discussion on BCO model is presented in Section IV. Finally, this paper ends with conclusions and future works.
II. TRAVELLING SALESMAN PROBLEM
Assume that a salesman is given a set of cities along with traveling distances (or costs) from one city to another city. The salesman is obligatory to explore each city (while traveling the salesman must visit every city only once and then return to the starting city) with least distances (or costs). The outcome of TSP is to conclude a Hamiltonian tour with minimum cost. Generally, TSP can be denoted via graph notations as follows:
Let G = (V, E) be a graph. V is a set of x cities, V = {v1,
…, vx}. E is a set of arcs or edges, E = {(p,q ): p, q є V}.
E is usually coupled with a distance (or cost) matrix, which is defined as D = (d). If dp, q = dq, p, the problem is
a symmetric TSP (STSP) else it belongs to asymmetric TSP (ATSP). In TSP, the purpose is to establish the permutation π of set V that minimizes the total roundtrip distance as shown in Equation (1). Overview and fundamental concepts about TSP can be referred from [5], [6].
a-1
C(π)= ∑ dπ(j),π(j+1) + dπ(a),π(1) (1)
j=1
Various techniques used to solve TSP.
In general, these techniques are classified into two broad categories:
First are Exact and the other one is Approximation algorithms [2].
A. Exact algorithms
Exact algorithms are methods which make use of mathematical models while approximation algorithms make use of heuristics and iterative improvements for problem solving process. Some examples of exact methods category are Lagrangian Relaxation, Branch and Bound, Integer Linear Programming etc.
B. approximation algorithms
The approximation algorithms are of two types: 1) Constructive heuristics
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 5, May 2012) A greedy algorithm is an algorithm that follows the problem solving heuristic of making the locally optimal choice at each stage with the hope of finding a global optimum. On some problems, a greedy strategy need not produce an optimal solution, but nonetheless a greedy heuristic may yield locally optimal solutions that approximate a global optimal solution.
The nearest neighbor algorithm was one of the first algorithms used to determine a solution to the travelling salesman problem. In it, the salesman starts at a random city and repeatedly visits the nearest city until all cities have been visited. It quickly yields a short tour, but usually not the optimal one.
The insertion heuristic algorithm gradually builds a route by inserting at each step a vertex in its neighborhood on the current route, and performing a local reoptimization. This is done while checking the feasibility of the remaining part of the route. Backtracking is sometimes necessary. Once a feasible route has been determined, an attempt is made to improve it by applying a post-optimization phase based on the successive removal and reinsertion of all vertices.
The goal of the Christofides heuristic algorithm (named after Nicos Christofides) is to find a solution to the instances of the traveling salesman problem where the edge weights satisfy the triangle inequality.
2) Improvement heuristics.
Instances in improvement heuristic include k-opt [5], Lin-Kernighan Heuristics [6] [7] [23], Simulated Annealing [8], Tabu Search [9], Evolutionary Algorithms [10] [11] [12], Ant Colony Optimization [13] [14], Bee System [15] etc.
In combinatorial optimization, Lin–Kernighan is one of the best heuristics for solving the Euclidean traveling salesman problem. It briefly involves swapping pairs of sub-tours to make a new tour. It is a generalization of 2-opt and 3-2-opt. 2-2-opt and 3-2-opt work by switching two or three paths to make the tour shorter. Lin–Kernighan is adaptive and at each step decides how many paths between cities need to be switched to find a shorter tour.
Simulated annealing (SA) is a generic probabilistic metaheuristic for the global optimization problem of locating a good approximation to the global optimum of a given function in a large search space. It is often used when the search space is discrete (e.g., all tours that visit a given set of cities). For certain problems, simulated annealing may be more efficient than exhaustive enumeration — provided that the goal is merely to find an acceptably good solution in a fixed amount of time, rather than the best possible solution.
Tabu search is a local search method used for mathematical optimization. Local searches take a potential solution to a problem and check its immediate neighbors (that is, solutions that are similar except for one or two minor details) in the hope of finding an
improved solution. Local search methods have a tendency to become stuck in suboptimal regions or on plateaus where many solutions are equally fit. Tabu search enhances the performance of these techniques by using memory structures that describe the visited solutions or user-provided sets of rules. If a potential solution has been previously visited within a certain short-term period or if it has violated a rule, it is marked as "taboo" so that the algorithm does not consider that possibility repeatedly.
The ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. The algorithm aims to search for an optimal path in a graph, based on the behavior of ants seeking a path between their colony and a source of food. Researchers focus on TSP from various disciplines because of its easy initiation, complicated to solve but possess numerous applications.
III. LITERATURE SURVEY
Because of many similarities between server and honey bee colony forager allotment, [1] proposed a new decentralized honey bee algorithm which with dynamism allocates servers to give surety of request loads. The work is compared in opposition to an omniscient optimality algorithm, a conventional greedy algorithm, and an algorithm that computes omnisciently the optimal static allocation. They evaluated performance on simulated request streams and commercial trace data. The algorithm shows better result than static or greedy for highly variable request loads, but greedy can outperform the method under low changeability.
A number of results were developed, by Chandra and Tovey [5] some worst-case and some probabilistic, on the performance of 2- and k-opt local search for the TSP, with respect to both the quality of the solution and the speed with which it is obtained.
An implementation of the Lin-Kernighan heuristic, one of the most effective methods for giving optimal or near optimal solutions for the symmetric TSP is described [7], [13] Computational tests show that the implementation is highly valuable.
Vanlaarhoven [8] offered a quantitative study of the typical behavior of the simulated annealing algorithm based on a cooling schedule obtainable previously by the authors. The study is based on the analysis of numerical results obtained by systematically applying the algorithm to a 100-city TSP. The prospect and the variance of the cost are analyzed as a function of the control parameter of the cooling schedule
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 5, May 2012)311 Tabu search and its application to the symmetric TSP, which is a classic combinatorial optimization problem is shown by Knox [9]The results of Tabu search is compared to the K-OPT procedure using six test problems drawn from the literature
The combination of local search heuristics and genetic algorithms is a capable approach for finding near-optimum solutions to the traveling salesman problem (TSP). An approach is offered in which local search techniques are used to find local optima in a given TSP search space, and genetic algorithms are used to search the space of local optima in order to find the global optimum. In this Merz utilized[10] new genetic operators for realizing the proposed approach are described, and the quality and efficiency of the solutions obtained for a set of symmetric and asymmetric TSP instances are discussed.
White and yen give details the development of a Hybrid Evolutionary Algorithm for solving the Traveling Salesman Problem (TSP).[12] The stratagem of the algorithm is to broaden the successful results of a genetic algorithm (GA) using a distance preserving crossover (DPX) by including memory in the form of ant pheromone during the city selection process. The synergistic combination of the DPX-GA with city selection based on probability found out by both distance and previous success adds additional information into the search mechanism. This combination into a Hybrid GA facilitates finding quality solutions for TSP problems with lower computation complexity.
MAX --MIN Ant System, an enhanced version of basic Ant System, and report the results for its application to symmetric and asymmetric instances of Traveling Salesman Problem. Hoos [14] show how MAX --MIN Ant System can be significantly refined extending it with local search heuristics. The results show that MAX --MIN Ant System has the property of effectively guiding the local search heuristics towards promising regions of the search space by generating good initial tours.
Seeley et al. [18] studied the extent to which scout honey bees gain information from waggle dances the whole time their careers as foragers.
Swarm intelligence is a research branch that models the population of interacting agents or swarms that are able to self-organize. An ant colony, a flock of birds or an immune system is a typical example of a swarm system. Bees’ swarming around their hive is also an example of swarm intelligence. Artificial Bee Colony (ABC) The approach proposed by Karaboga and Basturk [19] is an optimization algorithm based on the intellectual behavior of honey bee swarm.
In this work, ABC algorithm is used for optimizing multivariable functions and the results produced by ABC, Genetic Algorithm (GA), Particle Swarm Algorithm (PSO) and Particle Swarm Inspired Evolutionary
Algorithm (PS-EA) have been compared. The results showed that ABC outperforms the other algorithms. Traditionally, chess has been called the "fruit fly of Artificial Intelligence". In recent years, however, the focus of game playing research has gradually shifted away from chess towards games that offer new challenges highlighted by Rijswijk [20]. These challenges consist of large branching factors and imperfect information. The key ideas about Queen Bee are presented in the approach.
Chong and Gay [21] presented a novel approach that uses the honey bees foraging model to solve the job shop scheduling problem.
Chong and Gay proposed a population-based approach that uses a honey bees foraging model to solve job shop scheduling problems. [22]The algorithm uses an efficient neighborhood structure to search for reasonable solutions and iteratively progress on prior solutions. The primary solutions are generated using a set of priority dispatching rules. Experimental results comparing the proposed honey bee colony approach with existing approaches such as tabu search, ant colony and shifting bottleneck procedure on a set of job shop problems are presented. .
IV. THEBCOMODEL FOR TSP
Various bees models have been attempted in different domains ranging from dynamic server allocation for internet hosting center [1], numerical function optimization[19], hex game playing program [20], job shop scheduling [21] [22], to TSP.
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 5, May 2012) BCO model is a population based search algorithm mimics the food foraging behavior of swarms of honey bees [16]. The algorithm performs a kind of neighborhood search combined with random search. The basic approach behind the BCO is to produce the multi agent system (colony of artificial bees) skilled to successfully solve difficult combinatorial optimization problems. The artificial bee colony behaves partially alike, and partially differently from bee colonies in nature.
When a honeybee scout finds food, she uses two known tools to understand where it is. One is her solar compass, which lets her remember where things are in relation to the sun. [17][18]The bee's ability to see polarized light lets her determine where the sun is regardless of whether it is masked by clouds. The other tool is her internal clock, which lets her keep track of how far she has flown.
When she returns to the hive, the scout bee recruits her sisters to carry the food back to the nest. They, like the scout, are the oldest bees in the hive. The scout distributes samples of the food, which will help her sisters find the food when they reach their destination. Then, she performs a dance on the vertical surface of the combs in the hive. The area on which she performs the dance is commonly known as the dance floor, and the worker bees who observe the dance are followers.
If the food is nearby, the bee performs a round dance by traveling in loops in alternating directions. The round dance doesn't convey much information about exactly where the food is. However, it's generally close enough that the worker bees can smell it fairly quickly.
When the food is far away, the scout performs a waggle dance. During the waggle dance, the scout runs in a straight line while waggling her abdomen, and then returns to the starting point by running in a curve to the left or right of the line [17] [18]. The straight line indicates the direction of the food in relation to the sun. If the bee runs straight up the hive wall, then the foragers can find the food by flying toward the sun. If she runs straight down the wall, then the foragers can find the food by flying away from the sun. As the dance progresses, the dancing bee adjusts the angle of the waggle run to match the movement of the sun.
C. Path Construction in Bee
The scout bees after moving randomly generates two set of paths namely preferred path and possible path. Possible path set contains the cities that can be visited from the current city. The preferred path set contains the cities that results in more economic move in terms of cost as well as distance.
The state transition rule from one city to another is based on two parameters:
Arc fitness
Heuristic distance
A higher fitness value is assigned for a path which is a part of preferred path. Because of heuristic distance influence a bee tends to choose the next visiting city which is nearest to current city. State transition probability gives the likelihood to move from city i to city j after n transitions. It is a function of the distance between cities and of the arc fitness present on connecting edge.
Pij (t) = [ρij (t) α
. [1/dij] β
(2)
Σ [ρ
ij(t)]α.[1/dij] β
j ϵ Ai (t)
Where ρij (t) is the arc fitness from city I to city j after n
transitions,
dij represents the distance between city i and city j,
α represents a binary variable that turns on or off the arc fitness influence in the model, and
β is to control the significant level of heuristic distance. The general procedure that is followed by bee for solving TSP is shown below:
Procedure BCO Initialize_Population ( )
While stop criteria are not fulfilled do
While all bees have not built a complete path do Observe_Dance ( )
Forage_ByTransRule ( ) Perform_Waggle_Dance ( ) End while
End while
End procedure BCO
The waggle dance performed by bees can be illustrated as follows in figure:
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 5, May 2012)313 In the above figure the bees communicate via the waggle dance, the dance gives the angle of the food source with respect to sun, here angle is denoted by α, and distance is shown by the long run which is 1 km here in the figure.
V. CONCLUSION AND FUTURE WORK
This paper presents a glance of various approaches being used to solve TSP. The Bee Colony Optimization is highlighted for Travelling Salesman problem with its basic mechanism of bees foraging behavior and its efficiency in solving shortest path among various routes. Among the future enhancements are incorporating neighborhood search in the model. Neighborhood search is useful when exploitation is desired. It can be applied after every bee cycle to enhance the quality of solutions. BCO can be combined with other heuristic techniques such as k-opt local search, Tabu search to get more fine results. Constructive heuristics can also be included in preparing preliminary solutions. This helps the BCO algorithm to have a enhanced starting point in its search for finest solutions. More bees’ related features such as the direction shown in waggle dance and scent sensing can be included to make the model broader.
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International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 5, May 2012)[19] D. Karaboga and B. Basturk, "A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm," Journal of Global Optimization, vol. 39, no. 3, pp. 459-471, 2007.
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