## 5 Experiments and Results
As an example, the standard deviation for benchmark instance p09 is 3.047, which is 0.22% of the solution. The mean of the average deviation values is 0.41% and of the standard deviation values 0.55%. The principle of estimating the quality of solutions by computing the standard deviation is shown in Table 5, which compares these values for the BKS solutions (see Table 3) and solutions obtained by the original ACO algorithm and the ACO algorithm with AOP (see Table 2). As can be seen, the values of the standard deviation are higher in most cases, for worse solutions. The mean of the standard deviation for the BKS solutions is 10.98%, 2.20% for the solutions of the original ACO, and 0.55% for the solutions of the ACO with AOP. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Instance** & **Routes (Solution in Bold)** & **AvgDev (\%)** & **StDev (\%)** \\ \hline p01 & 152.24 151.86 151.02 **152.54** & 0.477 (0.31\%) & 0.661 (0.43\%) \\ p02 & 125.15 1125.47 **125.82** 118.90 & 2.469 (1.96\%) & 3.303 (2.63\%) \\ p03 & 133.27 133.52 134.14 133.84 **136.05** & 0.754 (0.55\%) & 1.104 (0.81\%) \\ p04 & **511.41** 509.95 & 0.732 (0.14\%) & 1.035 (0.20\%) \\ p05 & **378.17** 372.55 & 2.808 (0.74\%) & 3.971 (1.05\%) \\ p06 & 301.02 300.141 **301.75** & 0.551 (0.18\%) & 0.802 (0.27\%) \\ p07 & **228.95** 1224.81 226.79 1227.00 & 1.088 (0.48\%) & 1.695 (0.74\%) \\ p08 & 2224.81 **2224.85** & 0.020 (0.00\%) & 0.028 (0.00\%) \\ p09 & **1364.70** 1362.17 1358.63 & 2.134 (0.16\%) & 3.047 (0.22\%) \\ p10 & 957.14 1960.76 194.97 14.91 958.09 & 3.347 (0.35\%) & 4.720 (0.49\%) \\ p11 & 741.42 174.47 **491.41** 748.34 1745.70 & 2.328 (0.31\%) & 3.133 (0.42\%) \\ p12 & **659.48** 1659.48 & 0.000 (0.00\%) & 0.000 (0.00\%) \\ p13 & **659.48** 1659.48 & 0.000 (0.00\%) & 0.000 (0.00\%) \\ p14 & **682.84** 682.84 & 0.000 (0.00\%) & 0.000 (0.00\%) \\ p15 & 638.15 640.58 636.12 1 639.28 & 1.398 (0.22\%) & 1.891 (0.30\%) \\ p16 & **651.55** 651.55 61.55 & 651.55 & 0.000 (0.00\%) & 0.000 (0.00\%) \\ p17 & **677.27** 677.27 677.27 677.27 & 0.000 (0.00\%) & 0.000 (0.00\%) \\ p18 & 646.11 641.43 647.94 **652.24** 650.51 632.78 & 5.375 (0.82\%) & 7.135 (1.09\%) \\ p19 & 645.53 643.08 650.47 649.13 **651.55** 645.04 & 2.917 (0.45\%) & 3.387 (0.52\%) \\ p20 & 67.85 677.27 677.27 **681.13** 677.27 677.27 & 2.831 (0.42\%) & 4.437 (0.65\%) \\ p21 & **659.25** 1638.74 1635.47 1637.61 644.01 653.75 1644.89 1655.23 655.63 & 7.814 (1.19\%) & 8.948 (1.36\%) \\ p22 & 653.49 1656.33 655.88 1643.55 1643.08 1657.90 1652.34 1654.27 656.57 & 4.185 (0.64\%) & 5.530 (0.84\%) \\ p23 & 675.58 1683.13 677.27 679.281 673.42 **686.69** 681.15 1683.13 1685.00 & 3.671 (0.53\%) & 4.460 (0.65\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of lengths of routes for solutions of benchmark instances. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Instance** & **BKS** & **Original ACO** & **ACO with AOP** \\  & **StDev (\%)** & **StDev (\%)** & **StDev (\%)** \\ \hline p01 & 75.912 (31.92\%) & 3.547 (2.25\%) & 0.661 (0.43\%) \\ p02 & 35.989 (21.22\%) & 3.040 (2.36\%) & 3.303 (2.63\%) \\ p03 & 50.421 (24.33\%) & 0.667 (0.48\%) & 1.104 (0.81\%) \\ p04 & 44.466 (8.35\%) & 0.743 (0.14\%) & 1.035 (0.20\%) \\ p05 & 4.457 (1.18\%) & 0.605 (0.16\%) & 3.971 (1.05\%) \\ p06 & 122.020 (28.21\%) & 2.761 (0.88\%) & 0.802 (0.27\%) \\ p07 & 21.018 (8.54\%) & 9.725 (3.92\%) & 1.695 (0.74\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of lengths of routes for various solutions.