## 4 Analysis
### Max-Flow Problems for Network Optimization
Step 3: Total Maximum Flow CalculationThe total maximum flow \(F\) is the sum of all flows from Dar es Salaam to each destination:  
\[F=\sum_{(s,t)\in F}fsi\]  
Defining capacities for each route means determining how much cargo each transportation path (like roads or railways) can handle daily. This involves assessing the maximum cargo flow that each route can accommodate without exceeding its limits as shown in Tables 5 and 6. Available capacity: 1300 tons/h. Cargo demand: 1000 t/day. Flow allocated: We allocate the full 1000 t/day. Dar es Salaam \(\rightarrow\) Lilongwe (via road):  
Available capacity: 1200 tons/h. Cargo demand: 1200 t/day. Flow allocated: We allocate the full 1200 t/day. Dar es Salaam \(\rightarrow\) Lubumbashi (via road):  
Available capacity: 2000 tons/h. Cargo demand: 2500 t/day. Flow allocated: We can only allocate 2000 t/day due to the road capacity limit. 2. Residual capacities and adjustments  
After the initial flow allocations, we check for any residual capacities on the routes and whether any adjustments can be made. The road to Lusaka has 1500 t/day allocated via rail and 300 t/day via road, fully utilizing Lusaka's demand. The total flow capacity utilized and any bottlenecks are checked and summarized as shown in Table 7.  
Step 5: Final Flow Allocation Results  
Max-Flow Summary  
The max-flow analysis shows that Dar es Salaam Port can transport 9700 t/day to hinterland cities, leveraging current road and rail infrastructure. While most routes meet demand, notable bottlenecks exist. Lusaka relies on a fully utilized rail link (1500 t/day) supplemented by 300 t/day via road, highlighting the rail's critical role in meeting its 1800-ton/day demand. Conversely, Lubumbashi faces a 500-ton/day shortfall, as road capacity (2000 t/day) cannot accommodate its 2500-ton/day demand, emphasizing the need for infrastructure expansion or alternative transport options. Other routes display varying capacity utilization. Kampala's road infrastructure matches its demand (2500 t/day), operating at full capacity with no room for growth. Kigali and Bujumbura have excess capacities of 300 t/day, supporting future demand increases or surges. However, Lilongwe operates at its full 1200-ton/day road capacity, necessitating upgrades for any additional growth. Addressing these challenges, particularly for Lubumbashi, is crucial to optimizing cargo flow and ensuring seamless regional trade connectivity. Key Observations:  
The Lusaka rail line is vital for meeting the city's 1800-ton/day cargo demand, with 1500 t/day handled by rail and the remaining 300 t/day by road. This dual transport mode prevents the road infrastructure from becoming overwhelmed, relying solely on roads would strain its capacity. Conversely, most road networks, like those to Kigali and Bujumbura, have 300 t/day of unused capacity, providing flexibility to accommodate future demand growth. However, the road to Lubumbashi faces a 500-ton/day shortfall, creating a significant bottleneck that hampers full cargo delivery and highlights the urgent need for infrastructure upgrades or alternative transport solutions, such as rail expansion. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Rout** & \multicolumn{1}{c}{**Mode of Transport**} & **Cargo Flow (Tons/Day)** & **Unused Capacity** \\ \hline Dar es Salaam \(\rightarrow\) Kampala & Road & 2500 t/day & 57,500 t/day \\ \hline Dar es Salaam \(\rightarrow\) Kigali & Road & 1200 t/day & 34,800 t/day \\ \hline Dar es Salaam \(\rightarrow\) Lusaka & Road & 300 t/day & 42,900 t/day \\ \hline Dar es Salaam \(\rightarrow\) Lusaka & Rail & 1500 t/day & 0 t/day \\ \hline Dar es Salaam \(\rightarrow\) Bujumbura & Road & 1000 t/day & 30,200 t/day \\ \hline Dar es Salaam \(\rightarrow\) Lilongwe & Road & 1200 t/day & 27,600 t/day \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The final flow allocation for each route. The analysis underscores the importance of targeted infrastructure planning.