# Improvement of Network Flow Using Multi-Commodity Flow Problem
## 6 Discussion  
### Objective Function Using the Maximin Principle  
The proposed method can prevent one commodity from having zero flow and no routing. This makes it easy to apply to real networks. Adding \(v\) to the objective function does not necessarily give the maximum flow of the entire network. The total flow in Figure 8 is almost equal to the maximum flow in Figure 6. Therefore, in this example, there is almost no change in network utilization even if the objective function is changed. In Figure 4, where there are only two links, the total flow is now 1 less than the maximum flow, so we can say that the network utilization has decreased. However, we think that the more links there are on the network, the more alternative paths exist that get closer to the maximum flow, so the actual network does not open up a large difference in its utilization. The maximum flow can be obtained by setting the value of the weight \(w_{1}\) of \(v\) to 0.  
If flow bias occurs between commodities, the commodity with the small flow may be a single path. If the single path differs from the optimal path selected by OSPF, the packet arrival rate and average end-to-end delay for that commodity will be worse than OSPF. The proposed method satisfies the concept of Max-Min fairness and emphasizes uniform allocation. This makes it easier to prepare multiple paths for each commodity,  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **OSPF-ECMP** & **Proposed Method** \\ \hline Commodity 1 & 37 ms & 43 ms \\ \hline Commodity 2 & 39 ms & 55 ms \\ \hline Commodity 3 & 28 ms & 29 ms \\ \hline \hline \end{tabular}
\end{table}
Table 6: Average end-to-end delay for 3 sources and 3 destinations.  
Figure 11: Packet arrival rate for 3 sources and 3 destinations.  
improving the throughput of the entire network. In particular, commodities that benefit from the proposed method are those whose flows are smaller with the original calculation method. In Figure 7, the packet arrival rate decreased considerably because the flow of commodity 3 was small and had a single path. In Figure 9, using the proposed method, the arrival rate was higher due to the increased number of paths. Conversely, commodities that can monopolize the flow with the original calculation method have smaller flows with the proposed method, which may degrade the performance of load balancing. Comparing commodity 1 in the same figure, Figure 9 has a lower reachability rate than Figure 7 because the flow is smaller due to the emphasis on fairness.  
In the proposed method, by adding \(v\) to the objective function, there was vertex where the objective function can be maximized rather than simply calculating the maximum flow. In this case, the values of \(w_{1}\) and \(w_{2}\) were set to 0.5. In examples where there is a significant difference in the bandwidth of each link, the objective function may be maximized even for small values of \(v\). In that case, the value of \(v\) can be increased by increasing the value of \(w_{1}\). In general, as the value of \(w_{1}\) is increased, the fairness among commodities increases, but the network utilization tends to decrease, as shown in Figure 4. However, there are examples where it remains almost the same, as in Figure 8.