## 5 Numerical Tests and Comparisons
### Computational and Statistical Evaluation  
To compare the proposed heuristic with the existing heuristics, we carried out the experimentation by running each instance independently 10 times; for each replication we used \"Average Relative Percentage Deviation\" (ARPD) as a performance measure, which is popular in the scheduling literature [9; 10; 14; 16; 17]. ARPD is given by:  
\[ARPD=\frac{100}{k}\sum_{i=1}^{k}\frac{(Heuristici-BestHi)}{BestHi} \tag{12}\]  
where \(Heuristic_{i}\) is the total flowtime obtained by any of four algorithms, and the \(BestH_{i}\) is the lowest total flowtime obtained for that specific instance. Table 3 displays a comparative evaluation of the proposed metaheuristic, F&V [3], DPSO [9], Pan-Ruiz [10], and HPSO [31] based on ARPD for the Taillard benchmark data suite for 500 jobs. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Instances** & **F\&V** & **DPSO** & **PAN-** & **HPSO** & **PHPSO** & **Instances** & **F\&V** & **DPSO** & **PAN-** & **HPSO** & **PHPSO** \\ \hline ta001 & 0.6602 & 0.6602 & 0.4864 & 0.4864 & 0 & ta062 & 0.3453 & 0.3243 & 0.0779 & 0.0779 & 0 \\ ta002 & 0.8378 & 0.8378 & 0.6142 & 0.6142 & 0 & ta063 & 0.341 & 0.3253 & 0.0802 & 0.0802 & 0 \\ ta003 & 0.3002 & 0.3002 & 0.0931 & 0.0931 & 0 & ta064 & 0.3614 & 0.353 & 0.112 & 0.112 & 0 \\ ta004 & 0.5711 & 0.5711 & 0.3505 & 0.3505 & 0 & ta065 & 0.2568 & 0.2431 & 0.0316 & 0.0316 & 0 \\ ta005 & 0.6642 & 0.6642 & 0.4693 & 0.4699 & 0 & ta066 & 0.2517 & 0.2355 & 0 & 0 & 0.6682 \\ ta006 & 0.8226 & 0.8226 & 0.543 & 0.543 & 0 & ta067 & 0.41 & 0.399 & 0.1294 & 0.1294 & 0 \\ ta007 & 0.7412 & 0.7412 & 0.5032 & 0.5032 & 0 & ta068 & 0.2638 & 0.246 & 0 & 0 & 0.578 \\ ta008 & 0.2086 & 0.2086 & 0.0566 & 0.0566 & 0 & ta069 & 0.2751 & 0.2606 & 0.0276 & 0.0276 & 0 \\ ta009 & 0.6369 & 0.6369 & 0.4284 & 0.4284 & 0 & ta070 & 0.2429 & 0.2272 & 0 & 0 & 0.5659 \\ ta010 & 0.2729 & 0.2729 & 0.0747 & 0.0747 & 0 & ta071 & 0.5933 & 0.5818 & 0.152 & 0.1519 & 0 \\ ta011 & 0.449 & 0.449 & 0.2021 & 0.2021 & 0 & ta072 & 0.6913 & 0.6736 & 0.1762 & 0.1762 & 0 \\ ta012 & 1.4844 & 1.4844 & 1.1164 & 0 & ta073 & 0.6287 & 0.6143 & 0.1562 & 0.1562 & 0 \\ ta013 & 1.4634 & 1.4634 & 1.1326 & 1.1326 & 0 & ta074 & 0.6034 & 0.586 & 0.143 & 0.143 & 0 \\ ta014 & 0.4945 & 0.4945 & 0.2571 & 0 & ta075 & 0.7386 & 0.725 & 0.2368 & 0.2368 & 0 \\ ta015 & 1.2817 & 1.2817 & 0.8373 & 0.8373 & 0 & ta076 & 0.5632 & 0.5474 & 0.0772 & 0.0772 & 0 \\ ta016 & 1.7866 & 1.7866 & 1.4364 & 1.4364 & 0 & ta077 & 0.4528 & 0.4413 & 0.03 & 0.03 & 0 \\ ta017 & 1.8615 & 1.8615 & 1.3951 & 1.3951 & 0 & ta078 & 0.7447 & 0.7325 & 0.2625 & 0.2625 & 0 \\ ta018 & 0.9409 & 0.9409 & 0.6262 & 0.6262 & 0 & ta079 & 0.6651 & 0.6526 & 0.2063 & 0.2063 & 0 \\ ta019 & 0.6716 & 0.6716 & 0.446 & 0.446 & 0 & ta080 & 0.6151 & 0.987 & 0.1464 & 0.1464 & 0 \\ ta020 & 1.1856 & 1.1856 & 0.8944 & 0.8944 & 0 & ta081 & 1.2881 & 1.272 & 0.4875 & 0.4875 & 0 \\ ta021 & 3.459 & 3.459 & 2.8844 & 2.8844 & 0 & ta082 & 1.1395 & 1.1295 & 0.413 & 0.413 & 0 \\ ta022 & 3.0842 & 3.0842 & 2.4337 & 2.4337 & 0 & ta083 & 1.2602 & 1.2526 & 0.487 & 0.487 & 0 \\ ta023 & 2.7716 & 2.7716 & 2.3392 & 2.3392 & 0 & ta084 & 1.3522 & 1.3387 & 0.5561 & 0.5561 & 0 \\ ta024 & 2.4208 & 2.4208 & 1.7912 & 1.7912 & 0 & ta085 & 1.0442 & 1.0261 & 0.3556 & 0.3556 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of performance of the existing metaheuristics and PHPSO. Table 3 exhibits the results showing that the ARPD of PHPSO is significantly less compared to existing algorithms. It is also observed that, with respect to ARPD, the proposed method performs better than either of the existing methods for 103 problems out of the 120 under consideration. To validate the significance of the proposed algorithm statistically, the results of PHPSO are compared with the results obtained by earlier developed metaheuristics viz. F&V (2003), DPSO (2008), Pan-Ruiz (2012), and HPSO (2014).