*4.1. Parameter Setting*

In the following experiments, the parameters are setting as *m* = 50, *m*1 = 40, *m*2 = 10, *α* = 1, *β* = 2, *ρ*=0.05, *Q*=1, *N*max = 2000, *g* = 0.5 for ACO-MaxIt, *R*0 = 0.6 while *R* ∈ (0, 1) for ACO-Rand. Twenty cycles of experiments are carried out for each example independently. Then, minimum solution *S*min, maximum solution *S*max, average solution *Savg*, standard deviation *Std*, and average runtime *Tavg* for different examples of 20 times are given in the tables, where minimum solution *S*min, maximum solution *S*max, and average solution *Savg* are the percentage value deviation against the known optimal solution. The minimum value in each result is bolded in the tables.

#### *4.2. Experimental Results Comparison Based on AS*

First, we employ AS to three kinds of opposite based ACO, called AS-Index, AS-MaxIt, and AS-Rand, to verify the effectiveness of the improved algorithm. Twenty-six TSP examples are divided into three main categories, the small-scale, the medium-scale, and the large-scale according to the number of cities, respectively.

Small-scale city example sets are selected from TSPLIB, including eil51, st70, pr76, kroA100, eil101, bier127, pr136, pr152, u159, and rat195. The results are shown in Table 1.

From Table 1, the proposed AS-Index, AS-MaxIt, and AS-Rand show superior performances over AS for the examples, st70, kroA100, eil101, bier127, pr136, and u159. For other examples, the proposed algorithms outperform AS in general, except eil51. Meanwhile, stability by standard deviation is the not the primary concern when evaluating an algorithm. Compared among three proposed algorithms, AS-MaxIt illustrates superior performances for most cases.

To show more details in the process of evolutionary, curves for different stages are given in Figure 1 based on the case of kroA100.

According to Figure 1, AS shows faster convergence speed than the other three proposed methods in early iterations, while AS-Index, AS-MaxIt, and AS-Rand all surpass AS in average path length in later iterations. Meanwhile, AS-MaxIt performs best among all these algorithms, which also verifies the results in Table 1.

In the early stage, opposite path information introduced by OBL has negative impact on the convergence speed for all three proposed algorithms; however, it can provide extra information which guarantees the boost in accuracy for the later stage. The results lie in the fact that introducing extra information of opposite paths help to increase the diversity of the population, which balances the exploration and exploitation of solution space.

Medium-scale city example sets are selected from TSPLIB, including kroA200, ts225, tsp225, pr226, pr299, lin318, fl417, pr439, pcb442, and d493. The results are shown in Table 2.

From Table 2, it can be found that the proposed algorithms outperform AS in all the cases except ts225. Among all three algorithms, AS-Index and AS-MaxIt perform similarly, but better than AS-Rand generally. From these results, it can be seen that, with the help of extra information from opposite paths, three proposed methods all improve the original AS in solution accuracy.


**Table 1.** Results comparison for small-scale example sets.

Taking fl417 as the example, evolutionary curves in detail for different iteration stages are given in Figure 2, accordingly. According to Figure 2, AS also converges faster than the other three proposed methods in early iterations—for example before 1000 iterations. In addition, in later iterations, the other three proposed methods all exceed AS in average path length. This further validates the conclusions obtained from the analysis of Figure 1.

**Figure 1.** Evolutionary curves for different iteration periods based on kroA100.


**Table 2.** Results comparison for medium-scale example sets.

**Figure 2.** Evolutionary curves for different iteration periods based on fl417.

Large-scale city example sets are selected from TSPLIB, including att532, rat575, d657, u724, vm1084, and rl1304. The results are shown in Table 3.

From Table 3, it can be discovered that the AS-Index shows the obvious superior performance over all the other algorithms, which reveals a fact that the advantages of AS-Index appears as the scale of the example increases based on these results.

Taking vm1084 as the example, evolutionary curves in detail for different iteration stages are given in Figure 3, accordingly. According to Figure 3, AS still shows faster convergence speed than the other three proposed methods in early iterations, but AS-Index outperforms all the others in the end.

Based on all the tables and figures, it can be found that, in most scenarios, at least one of AS-Index, AS-MaxIt, and AS-Rand outperforms AS in average path length. For small-scale examples, AS-MaxIt shows better performance, while, for medium-scale cases, AS-Index and AS-MaxIt perform similarly better than the others. For large-scale city sets, AS-Index is the best algorithm, while AS-Rand ranks in the middle for most cases regarding figures, and it illustrates its stability to some extent. Therefore, it can be drawn that the strategy to introduce OBL into AS provides more information, namely better exploration capability, which explains the superiority of these proposed methods over the original AS. By comparing the results of the running time from Tables 1–3, we can also find that the running time of the three improved algorithms is not significantly increased compared with AS. It also validates our previous discussion on time complexity.

#### *4.3. Experimental Results Comparison Based on PS-ACO*

To further verify the effectiveness of the proposed algorithm, we employed another PS-ACO to three kinds of opposite based ACO, PS-ACO-Index, PS-ACO-MaxIt, and PS-ACO-Rand to verify the effectiveness of the improved algorithm. The number of ants is 50, and the other parameters are the same as in [26]. Twelve sets of TSP examples are eil51, st70, kroA100, pr136, u159, rat195, tsp225, pr299, lin318, fl417, att532, and d657. The results are given in Table 4.

From Table 4, the proposed PS-ACO-Index, PS-ACO-MaxIt, and PS-ACO-Rand show superior performances over PS-ACO for the examples, eil51, st70, rat195, tsp225, and pr299. For other examples, the proposed algorithms outperform PS-ACO in general, except lin318 and fl417. Compared among three proposed algorithms, PS-ACO-Rand illustrates superior performances for most cases. By comparing the results of the running time, we can also find that the running time of the three improved algorithms is not significantly increased compared with PS-ACO.


**Table 3.** Results comparison for large-scale example sets.

**Figure 3.** Evolutionary curves for different iteration periods based on vm1084.


**Table 4.** Comparisons of PS-ACO, PS-ACO-Index, PS-ACO-MaxIt, and PS-ACO-Rand.
