A faster, more reliable solver of regular-grid TSPs: single value threshold accepting

Abstract

The main point of this paper is to provide a simple and efficient threshold value for the threshold accepting (TA) algorithm to attempt an optimal solution for the regular grid travelling salesman problems (TSPs) in a reliable way. This new algorithm is named the single value threshold accepting algorithm (SVTA). The number of the threshold value is one and its value is: $\left\lceil {\left( {2 - \sqrt 2 } \right)g*10,000} \right\rceil /10,000$ where g is the grid size and $\left\lceil x \right\rceil$ is the smallest integer not less than a real number x. For the regular-grid TSPs, g can be set as $1/\sqrt n$ where n is the problem size. It is shown empirically, with 100 independent simulations performed with 441 cities, in 16 different cases, that the SVTA is far superior to (at least five times faster on average than) the previous double threshold accepting (DTA) with respect to the speed of finding a global optimal reliably. Particularly in the case of the 441 cities, the proposed algorithm is at least 21 times faster and rises up to 96 times on average and optimistically 284 times faster than the previous one. Important insight is provided that reveals how the formula works and why it works successfully.

Appendix A

Tian et al. [22] compared six perturbation schemes for generating random permutation solution in the SA algorithm. The six types are as follows:

PS1::

Two adjacent terms are interchanged

PS2::

Two terms are interchanged

PS3::

A single term is moved

PS4::

Subsequent terms are moved

PS5::

Subsequent terms are reversed

PS6::

Subsequent terms are reversed

For each instance of the experimental example, 100 runs were performed with different initial random solutions. The performance of the algorithm with the permutation schemes is quantified as:

$${\rm{Optimal}}\;{\rm{Ratio}}\left( {\rm{\% }} \right){\rm{ = }}{{Num_{opt} } \over {Num}} \times 100,$$

(A1)

where Num _opt is the number of runs attained global optimum, Num is the total number of runs:

$${\rm{Offset}}\;{\rm{Global}}\;{\rm{Optimum}}\;\left( {\rm{\% }} \right){\rm{ = }}{{f\left( \bullet \right) - f_{opt} } \over {f_{opt} }} \times 100,$$

(A2)

where f (●) is the best objective function value by a run, and f _opt is the exact global optimal value.

$${{\rm{Iter}}{\rm{.}}\;{\rm{Num}} = {\rm{the total iteration number of an evaluations}}{\rm{.}}}$$

(A3)

The results of the experimental evaluations for solving the regular grid TSPs with the SA algorithm with the perturbation schemes in the problem size of 16 through 100 cities are presented in Table 4. According to Tian et al., PS6 is the best of all schemes based on the abovementioned performance criteria. Based on the performance criteria of Tian et al., the performance of SVGA is also shown in Table 4. It is clear from Table 4 that SVTA is superior to Tian's perturbation schemes for all cases based on the criteria set up by Tian et al. The superiority of the SVTA over PS6 perturbation scheme in the SA is detailed as follows.

Table 4. The results of solving the regular-grid TSPs by the SA with the perturbation schemes and SVTA

Table 4 shows that except for the cases of both 16 and 100 cities, the SVTA always secures the global optimum within the specified iteration numbers. Thus, their optimal ratios are 100% and the means, variances and worst efforts of the offsets from the optimum are zeros. Whereas for the PS6 scheme, its optimal ratios are always less than 100% in the cases of 16 through 100 cities. In the 16 cities problem, the SVTA has an 99% optimal ratio, whereas the best of the six schemes, which is PS6, is 98%. The mean (0.0518) and variance (0.2681) of the SVTA are smaller than the ones (both 0.10 and 0.52 respectively) of PS6. In the worst case of the offset from the optimum, both the SVTA and PS6 are the same (5.18). Thus, the SVTA is better than PS6 in regard to the performance criteria of Tian et al. in the 16 cities problem. Meanwhile, in the 100 cities problem, the SVTA has a 98% optimal ratio, whereas the best of the six schemes, which is PS6, is 73%. The mean (0.0166) and variance (0.0136) of the SVTA are smaller than those (both 0.22 and 0.14 respectively) of PS6. In the worst case, both the SVTA and PS6 are the same (0.83). Thus, the SVTA is better than PS6 in regards to the performance criteria of Tian et al. in the 100 cities problem as well.