A company managing a railway network wishes to plan the repair of certain sections of its network which is represented by the graph below. The goods leave from the extraction point s and must reach the port at t. Each rail link has a capacity in terms of the number of trains that can run through it each period. The graph below indicates the capacity of each link per period.

Engineers have determined that the work in the table below must be performed. These must be completed within the next 4 periods. Some jobs can be done at any time, while others are constrained to certain times for operational reasons. Repairing a link requires reducing its capacity to perform the work. For each link, we will find the capacity of it when work is taking place.

The company must deliver as much cargo as possible to its port and must plan repairs to maximize the number of trains that can depart from s and get to t. has)

(25 points) Solve the problem with AMPL. Print the AMPL model and indicate in your report the paths taken for each period. Tip: do the maximum flow model in AMPL for one period first and make sure it works.

\begin{tabular}{c|c|c|c|c} \hline Arc & Duration & Periods & Capacity & Reduced capacity \\ \hline(s,1) & 1 & 1 ou 2 & 6 & 2 \\ (s,p) & 2 & & 8 & 4 \\ (1,2) & 1 & & 4 & 3 \\ (1,3) & 1 & 2 ou 3 & 4 & 0 \\ (1,5) & 2 & & 6 & 3 \\ (3,8) & 2 & & 8 & 0 \\ (5,7) & 2 & 2 ou 3 & 6 & 3 \\ (6,8) & 1 & 1 ou 4 & 4 & 1 \\ (7,8) & 2 & & 6 & 2 \\ (8,s) & 2 & & 6 & 4 \\ (8,t) & 1 & 1 ou 2 & 10 & 6 \\ \hline \hline \end{tabular} \begin{tabular}{c|c|c|c|c} \hline Arc & Duration & Periods & Capacity & Reduced capacity \\ \hline(s,1) & 1 & 1 ou 2 & 6 & 2 \\ (s,p) & 2 & & 8 & 4 \\ (1,2) & 1 & & 4 & 3 \\ (1,3) & 1 & 2 ou 3 & 4 & 0 \\ (1,5) & 2 & & 6 & 3 \\ (3,8) & 2 & & 8 & 0 \\ (5,7) & 2 & 2 ou 3 & 6 & 3 \\ (6,8) & 1 & 1 ou 4 & 4 & 1 \\ (7,8) & 2 & & 6 & 2 \\ (8,s) & 2 & & 6 & 4 \\ (8,t) & 1 & 1 ou 2 & 10 & 6 \\ \hline \hline \end{tabular}