## 3 Methods
Efficiency is a general term that, depending on the situation, can be defined and assessed in several ways. It frequently entails using the fewest resources--such as money, time, or energy--to achieve the intended result. When addressing problems or coming up with solutions, we typically take the amount of time needed to determine the efficiency of a particular approach. Efficiency, for instance, could be quantified as the ratio of the input--in this case, time--used to produce the output, which is the quality of the solution [16]. \[Efficiency\;(A)=\frac{Quality\;of\;Solution\;(A)}{Time\;consumed\;(A)} \tag{3}\] \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Instance** & **Time (ms)** & **Time \(\ast\) 2** & **GA Solution with Random** & **GA Solution with Initial Seeding with Time \(\ast\) 2** & **GA Solution with Initial** & **GA Solution with Initial** \\ \hline bier127 & 6023 & 12,046 & 214,928 & 187,452 & 128,678 & 139,460 \\ a280 & 19,399 & 38,798 & 8243 & 7148 & 3018 & 4525 \\ kroA100 & 5565 & 11,130 & 41,233 & 34,176 & 22,057 & 26,040 \\ kroA200 & 9403 & 18,806 & 96,272 & 75,012 & 32,088 & 43,140 \\ lin38 & 27,772 & 55,544 & 202,537 & 139,552 & 47,507 & 63,085 \\ pr144 & 6618 & 13,236 & 193,568 & 130,639 & 62,481 & 95,713 \\ att532 & 173,946 & 347,892 & 358,763 & 348,633 & 96,050 & 126,337 \\ att48 & 5136 & 10,272 & 46,780 & 40,711 & 34,877 & 38,544 \\ rat783 & 803,368 & 1,606,736 & 51,239 & 49,520 & 10,196 & 11,999 \\ pr439 & 89,720 & 179,440 & 380,537 & 390,937 & 122,586 & 166,083 \\ eil51 & 5151 & 10,302 & 493 & 477 & 438 & 463 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Computational results for the GA with a specific amount of time for each instance. Figure 7: Comparison of the final solution in a specific amount of time. where A is a problem-solving method, i.e., TSP, and the Quality of the Solution can be defined using Equation (2) (average convergence rate.) This allows the definition of efficiency for a TSP solver to be [16]

\[Efficiency\;(A)=\frac{Convergence\;Rate\;(A)}{Time\;consumed\;(A)} \tag{4}\]

This formula implies that a more efficient solution achieves a high-quality result using fewer resources (time). Table 6 and Figure 9 show the calculation of the error rate for each initial seeding method, but the termination criterion was running to the specified time. \begin{table}
\begin{tabular}{c c c c c} \hline
**Instance** & **GA with Random Seeding** & **GA with Random Seeding (Time \(\ast\) 2)** & **GA with IAMTSP+ Seeding** & **GA with Regression-Based Seeding** \\ \hline bier127 & 44.96669 & 36.9001131 & 8.0790811 & 15.18572 \\ a280 & 68.71285 & 63.91997762 & 14.546057 & 43.00552 \\ kroA100 & 48.386 & 37.72823034 & 3.5136238 & 18.27189 \\ kroA200 & 69.49476 & 60.84893084 & 8.4766891 & 31.92397 \\ lin318 & 79.24873 & 69.88291103 & 11.530932 & 33.37719 \\ pr144 & 69.75895 & 55.19178806 & 6.3123189 & 38.84112 \\ att532 & 92.28293 & 92.05869783 & 71.175429 & 78.0856 \\ att48 & 77.28089 & 73.89403355 & 69.527196 & 72.42632 \\ rat783 & 82.81387 & 82.21728595 & 13.632797 & 26.61055 \\ pr439 & 71.82482 & 72.57435341 & 12.537321 & 35.44372 \\ eil51 & 13.59026 & 10.6918239 & 2.739726 & 7.991361 \\ \hline \end{tabular}
\end{table} 
Table 6: The error rate for the GA with 3 initial seeding methods based on the specified time. Figure 8: The efficiency of the GA for bier127 from the TSPLIB using two population seeding methods: random and IAMTSP+. Based on Equation (3), Table 7 displays the GA efficiency for each instance, as well as the methods used to seed the initial population. From Table 7, we conclude that employing IAMTSP+ as an initial population seeding method with the GA improves the efficiency of the GA when solving the TSP. Table 7 demonstrates that the proposed GA-IAMTSP+ outperforms both the pure GA and the GA that initially began with a regression approach in terms of efficiency across all TSP instances. This notable achievement is attributable to the high-quality initial solutions offered by IAMTSP+.