Bachelor of Science

The Travelling Salesman Problem (TSP) is one of the classic problems in combinatorial optimisation. Its applications arise in problems like the delivery services, customer pick-up services, circuit board drilling, and even in X-ray cryxtallography experiments. As most of these applications involve an open tour, we shall therefore discuss the TSP as one involving an open tour.

TSP basically concerns finding the shortest total distance possible. Since the search for the shortest total distance is not easy to obtain especially when the number of cities is large, lower and upper bounds are introduced to at least find a near-optimal or an optimal solution.

In this paper, we used the Prim's algorithm and a particular heuristic algorithm to compute the lower and upper bounds respectively. We coded these algorithms on FORTRAN 77 so that we could investigate how much the upper bound varies fiom the lower bound. By taking the ratio of the upper bound to the lower bound, we could conduct an investigation on the probability distribution of the percentage of the upper bound exceeding the lower bound for different sets of cities.

Our first attempt at computing the upper and lower bounds shows that the former is on average about 30 percent above the latter, while for certain set of cities it maybe as high as 45 percent above. This percentage indicates that we need to improve the lower bound andlor the upper bound. The implementation of the sub-gadient optimisation method on the lower bound shows a relatively good improvement to 20 percent (maximum 30 percent). However, this can be further improved using a sub-case of the 3-opt algorithm on the upper bound to about 8 percent (maximum 16 percent). Though this 8 percent illustrates that the TSP solution has si@cantly improved, fiuther improvements can still be made by using the 3-opt algorithm or some other heuristic algorithms.