A Reliable Solver of Regular-Grid Travelling Salesman Problems: Double Threshold Accepting for Schedule Planning Optimisation with Unified Pattern of Normalised Threshold Values

Abstract

The threshold accepting (TA) method is a meta-heuristic algorithm to solve optimisation problems. Like the simulatedannealing (SA) method, the result of TA does not guarantee to converge to the global optimum within the truncated computer time. A double TA is proposed to overcome this problem and is applied to regular-grid travelling salesman problems (TSPs). It is discovered that with a proper selection of threshold values in a double TA, the empirical result shows that it always converges to the optimal solution in the regular-grid TSPs. The experiment is performed on 18 cases ranging from 16 to 441 points (cities) and each case is executed 100 times. They all converge to the optimal solution within the specified limit of the number of function evaluations. The pattern of the normalised threshold values is identical for all 18 cases. The pattern is in the shape of a natural cubic spline and can be constructed by eight control points. The threshold values in 18 cases are classified into three groups. In each particular group, although the pattern and size of the normalised threshold values is the same, the scale of the pattern is different for each case in the group. The scale is termed cap and can be obtained from the cap function of positive changes of the cost function in the steps of moves from a suboptimal tour to a global optimal tour. Empirically, our method is shown to be superior to the simulated annealing method in terms of the reliability of obtaining a global optimum within the truncated computer time.