A Comparative Study of Adaptive Crossover Operators for Genetic Algorithms to Resolve the Traveling Salesman Problem

International Journal of Computer Applications (0975 – 8887) Volume 31– No.11, October 2011 A Comparative Study of Adaptive Crossover Operators for Genetic Algorithms to Resolve the Traveling Salesman Problem ABDOUN Otman ABOUCHABAKA Jaafar LaRIT, Department of Computer Science LaRIT, Department of Computer Science IBN Tofail University, Kenitra, Morocco IBN Tofail University, Kenitra, Morocco ABSTRACT the modeler not only choose the representation methodology, Genetic algorithm includes some parameters that should be but also the cost function that judges the model’s soundness. adjusting so that the algorithm can provide positive results. The genetic algorithm is a one of the family of evolutionary Crossover operators play very important role by constructing algorithms. The population of a genetic algorithm (GA) evolves competitive Genetic Algorithms (GAs). In this paper, the basic by using genetic operators inspired by the evolutionary in conceptual features and specific characteristics of various biology, "The survival is the individual most suitable to the crossover operators in the context of the Traveling Salesman environment". Darwin discovered that species evolution based Problem (TSP) are discussed. The results of experimental on two components: the selection and reproduction. The comparison of more than six different crossover operators for selection provides a reproduction of the strongest and more the TSP are presented. The experiment results show that OX robust individuals, while the reproduction is a phase in which operator enables to achieve a better solutions than other the evolution run. operators tested. Genetic algorithms are powerful methods of optimization used successfully in different problems. Their performance is Keywords depending on the encoding scheme and the choice of genetic Travelers Salesman Problem, Genetic Algorithm, NP-Hard operators especially, the selection, crossover and mutation Problem, Crossover Operator, probability of crossover, Genetic operators. A variety of these latest operators have been Algorithm, suggested in the previous researches. In particular, several crossover operators have been developed and adapted to the 1. INTRODUCTION permutation presentations that can be used in a large variety of This section introduces the current scientific understanding of combinatorial optimization problems. In this area, a typical the natural selection process with the purpose of gaining an example of the most studied problems is the Traveling Salesman insight into the construction, application, and terminology of Problem (TSP). genetic algorithms. Natural selection –evolution- is discussed in many texts and treatises, and one of its first proponents, Charles The traveling salesman problem (TSP) is a classical problem of Darwin.His theory of evolution was based on four primary combinatorial optimization of Operations Research’s area. The premises [7]. First, like begets like; equivalently, an offspring purpose is to find a minimum total cost Hamiltonian cycle [22]. has many of the characteristics of its parents. This premise There are several practical uses for this problem, such as vehicle implies that the population is stable. Second, there are variations routing (with the additional constraints of vehicle’s route, such in characteristics between individuals that can be passed from as capacity’s vehicles) [23] and drilling problems [24]. one generation to the next. The third premise is that only a small percentage of the offspring produced survive to adulthood. The TSP has received considerable attention over the last two Finally, which of the offspring survive depends on their decades and various approaches are proposed to solve the inherited characteristics. These premises combine to produce the problem, such as branch-and-bound [28], cutting planes [35], 2- theory of natural selection. In modern evolutionary theory an opt [33], simulated annealing [31], neural network [1,37], and understanding of genetics adds impetus to the explanation of the tabu search [9, 29]. Some of these methods are exact algorithms, stages of natural selection. while the others are near-optimal or approximate algorithms. Another set of biologically-inspired methods are Genetic The exact algorithms include the integer linear programming Algorithms (GAs). They derive their inspiration from combining approaches with additional linear constraints to eliminate the concept of genetic recombination with the theory of infeasible subtours [25, 27, 30, 34, 36,36]. On the other hand, evolution and survival of the fittest members of a population [5]. network models yield appropriate methods that are flexible Starting from a random set of candidate parameters, the learning enough to include the precedence constraints [28,32]. More process devises better and better approximations to the optimal recently, genetic algorithm (GA) approaches are successfully parameters. The GA is primarily a search and optimization implemented to the TSP [26]. Potvin [35] presents survey of GA technique. One can, however, pose nearly any practical problem approaches for the general TSP. as one of optimization, including many environmental modeling These researches have provided the birth of several genetic problems. To configure a problem for GA solution requires that mechanisms in particular, the selection, crossover and the 49 International Journal of Computer Applications (0975 – 8887) Volume 31– No.11, October 2011 mutation operators. In order to resolve the TSP problem, we The mathematical formulation of TSP problem expresses by: propose in this paper to study empirically the impact affiliation of the different crossover operators.Finally we analyze the 푚푖푛{푓 푇 , 푇 = 푇 1 , 푇 2 , … … , 푇 푛 } (4) experimental results. 2. TRAVELING SALESMAN PROBLEM Which T[i] is a permutation on the set {1, 2, …,V}. The Traveling Salesman Problem (TSP) is one of the most intensively studied problems in computational mathematics.In The travelling salesman problem (TSP) is an NP-hard problem the TSP problem, which is closely related to the Hamiltonian in combinatorial optimization studied in operations research and cycle problem, a salesman must visit n cities. Modeling the theoretical computer science [5]. problem as a complete graph with n vertices, we can say that the Theorem: The subset-sum problem is NP-complete [3]. salesman wishes to make a tour, or Hamiltonian cycle, visiting each city exactly once and finishing at the city he starts from [1]. Proof :We first show that TSP belongs to NP. Given an instance Given the cost of travel between all cities, how should he plan of the problem, we use as a certificate the sequence of n vertices his itinerary for minimum total cost of the entire tour? in the tour. The verification algorithm checks that this sequence contains each vertex exactly once, sums up the edge costs, and As a concrete example, consider a delivery company with a checks whether the sum is at most k. This process can certainly central depot. Each day, it loads up each delivery truck at the be done in polynomial time. depot and sends it around to deliver goods to several addresses. At the end of the day, each truck must end up back at the To prove that TSP is NP-hard, we show that HAM-CYCLE ≤ P depotso that it is ready to be loaded for the next day. To reduce TSP. Let G =(V, E)be an instance of HAM-CYCLE. We costs, the company wants to select an order of delivery stops that construct an instance of TSP asfollows. We form the complete yields the lowest overall distance traveled by each truck. This graph G’ = (V, E’), , whereE’={(i,j) : i, j V andi ≠j }, and we problem is the well-known “Traveling Salesman Problem,” andit define the cost function c by is NP-complete [1]. It has no known efficient algorithm. Under 0 푖푓 푖, 푗 퐸 certain assumptions, however, we know of efficient algorithms 푐 푖, 푗 = (5) that give an overall distance which is not too far above the 1 푖푓 푖, 푗 퐸 smallest possible. (Note that because Gis undirected, it has no self-loops, and so The search space for the TSP is a set of permutations of n cities. c(v, v)=1 for all vertices vV.) The instance of TSP is then (G’, Any single permutation of n cities yields a solution (which is a c, 0), which we can easily create in polynomial time. complete tour of n cities). The optimal solution is a permutation We now show that graph Ghas a Hamiltonian cycle if and only which yields the minimum cost of the tour. The size of the if graphG’has atour of cost at most0. Suppose that graphGhas a search space is n!. Hamiltonian cycleh. Each edgeinhbelongs toE and thus has cost0 in G’. Thus,his a tour inG’with cost0. In other words, a TSP of size V is defined by a set of points v= Conversely, suppose that graphG’has a tourh’of cost at most0. {v1, v2, …,vn} which vi a city marked by coordinates vi.x and Since the costsof the edges inE’are0 and1, the cost of tourh’is vi.y where we define a metric distance function f as in (1). A exactly0and each edge on thetour must have cost0. solution of TSP problem is a form of scheduling Therefore,h’contains only edges inE. We conclude thath’is a T=(T[1],T[2],……,T[n], T[1]) which T[i] is a permutation on Hamiltonian cycle in graphG. the set {1, 2, …,V}. The evaluation function calculates the adaptation of each solution of the problem by the following A quick calculation shows that the complexity is O(n!) which n formula: is the number of cities (Table. 1). 푛−1 Table 1. Number of possibilities and calculation time by the 2 2 푓 = 푣푖. 푥 − 푣푖+1. 푥 + 푣푖. 푦 − 푣푖+1. 푦 number of cities 푖=1 Number of Number of Computation time cities possibilities + (푣 . 푥 − 푣 . 푥)2 + (푣 . 푦 − 푣 . 푦)2(1) 푛 1 푛 1 5 12 12 μs Where n is the number of cities.

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