# A New Modified Firefly Algorithm for Optimizing a Supply Chain Network Problem
## 3 Results and Discussion
### Experimental Results of Modified FAs  
In order to demonstrate the efficiency of the proposed modified FA, the performances of the developed modified FA was compared to the classical FA. Four performance measures were used in \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Problem** & **Problem** & **No. of** & **No. of** & **No. of** & **No. of** & **No. of** \\
**Size** & **Code** & **Plants (I)** & **DCs (J)** & **Retailers (K)** & **Items (P)** & **Periods (T)** \\ \hline \multirow{3}{*}{**Small**} & S1 & 2 & 2 & 4 & 2 & 6 \\  & S2 & 2 & 2 & 15 & 2 & 6 \\ \multirow{3}{*}{**Medium**} & M1 & 3 & 5 & 10 & 2 & 12 \\  & M2 & 2 & 8 & 15 & 4 & 6 \\ \multirow{3}{*}{**Large**} & L1 & 3 & 10 & 20 & 2 & 12 \\  & L2 & 6 & 12 & 25 & 8 & 12 \\ \cline{1-1}  & L3 & 8 & 15 & 40 & 8 & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Test problems size as suggested by [13]. this regard: (1) best fitness solution (Best_Sol), (2) CPU time, (3) relative percentage deviation (\(RPD\)) and (4) relative deviation index (\(RDI\)). \(RDI\) and the \(RPD\) are defined as ([PERSON], 2009):  
\[RDI=\left(\frac{Alg_{sol}-Min_{sol}}{Max_{sol}-Min_{sol}}\right)\times 100 \tag{13}\]  
\[RPD=\left(\frac{Alg_{sol}-Min_{sol}}{Min_{sol}}\right)\times 100 \tag{14}\]  
where \(Alg_{sol}\) is the value of the fitness function and \(Min_{sol}\) and \(Max_{sol}\) denote the best and the worst solutions, respectively. It must be noted that the lower values of both \(RDI\) and \(RPD\) show the better efficiency of the considered algorithm. Note also that in this section, the best out of 5 solutions obtained were selected for comparison. Table 3 provides a comparison study of classical FA and modified FA. As can be seen from the obtained results presented in Table 3, it is apparent the proposed modification makes the classical FA more efficient in terms of all considered performance metrics. The modified FA searches more in search space, therefore, it could obtain more feasible solutions. Closer inspection of the Table 3 reveals that the modified FA finds better solutions (in this case cheaper cost since the minimization problem is in run) for medium sizes problems (M1 and M2) by 1.69% and 1.51% respectively. The obtained solutions by modified FA (Best_Sol) are less than that of the classical FA by 0.41% on average for different sizes of problem. In addition, based on the required iterations to find the optimal solution, it can be seen that the modified algorithm can find the solution by 38.66% less iterations on average compared to the standard FA. Figure 5 confirms these claims. The results, as shown in Table 3, indicate that regarding required computational time, the modified FA performs more efficient compared to classical with a significant reduction of computational time by 8.12% on average for the different size of problems. From the results depicted in Figure 4, it is apparent that the reduction of computational time between modified FA and classical FA for medium sizes problems are 11.31% and 13.87%. A possible explanation for this might be that the convergence of classical FA slows down because of too much exploration in standard FA. In fact, standard FA explores all possible solutions including weak solutions in a search space. This exploration leads to increase the required computational time. As can be seen in the Figure 5, the best obtained solution function of both algorithms is almost identical. However, the modified algorithm found these values with less CPU time and fewer iterations. As a result, for more complicated and large sizes problems, the modified FA can perform more efficiently. In terms \(RPD\) metric while the values of metrics for both algorithms are close, however, for the smallest and the largest problem, the modified FA performs better than the standard FA. The values for \(RDI\) metric also show the modified FA in the smallest size problem \(S1\) performs significantly better than the standard FA.