1Population-Based Optimization Algorithms forSolving the Travelling Salesman ProblemMohammad Reza Bonyadi, Mostafa Rahimi Azghadiand Hamed Shah-HosseiniDepartment of Electrical and Computer Engineering,Shahid Beheshti University,Tehran, Iran1. IntroductionThe Travelling Salesman Problem or the TSP is a representative of a large class of problemsknown as combinatorial optimization problems. In the ordinary form of the TSP, a map ofcities is given to the salesman and he has to visit all the cities only once to complete a toursuch that the length of the tour is the shortest among all possible tours for this map. Thedata consist of weights assigned to the edges of a finite complete graph, and the objective isto find a Hamiltonian cycle, a cycle passing through all the vertices, of the graph whilehaving the minimum total weight. In the TSP context, Hamiltonian cycles are commonlycalled tours. For example, given the map shown in figure l, the lowest cost route would bethe one written (A, B, C, E, D, A), with the cost 31.Fig. 1. The tour with A=>B =>C =>E =>D => A is the optimal tour.In general, the TSP includes two different kinds, the Symmetric TSP and the AsymmetricTSP. In the symmetric form known as STSP there is only one way between two adjacentcities, i.e., the distance between cities A and B is equal to the distance between cities B and A(Fig. 1). But in the ATSP (Asymmetric TSP) there is not such symmetry and it is possible tohave two different costs or distances between two cities. Hence, the number of tours in theATSP and STSP onnvertices (cities) is(n-1)!and(n-1)!/2,respectively. Please note that thegraphs which represent these TSPs are complete graphs. In this chapter we mostly considerthe STSP. It is known that the TSP is an NP-hard problem (Garey & Johnson, 1979) and isoften used for testing the optimization algorithms. Finding Hamiltonian cycles or traveling
