(a) A company is planning to start $h$ new projects out of the $n$ project proposals. The selected projects must be started and finished within the next T months. Each project $i$ has an associated benefit denoted by $b_i$. The overall cost of completing the project $i$ is given by $c_i$ and the budget the company allocated for the selected projects is $\mathrm{C}$. To start project $i$ requires a set up time $s_i$ given in months and the number of months required to complete project $i$ is given by $d_i$. There are some restrictions in terms of which projects can be selected. Project 1 and 2 cannot be selected together. If project 3 is selected, then first project 4 and then project 5 must be completed before project 3 can be started. A project can be started at most once during the given $\mathrm{T}$ months and must be completed within the time horizon of T months. Formulate this problem so that the overall benefit for the company is maximized.
(b) The company is planning for the project portfolio to be realized within the next 24 months. There are 8 candidate projects and considering its management resources the company has decided to restrict its choice to 5 projects. A budget of $$\$ 1,840,000$$ has been allocated for the projects selected. The set-up time $\mathrm{s}_{\mathrm{i}}$ is given as 1 month for all candidate projects. The benefit ( $$\$ ' 000)$$, cost ($$\$'000)$$, and duration (months) for each candidate project are estimated as in the following table. The durations do not include the set-up time.
$$
\begin{array}{l|l|l|l|l|l|l|l|l}
\hline & \begin{array}{l}
\text { Project } \\
\# 1
\end{array} & \begin{array}{l}
\text { Project } \\
\# 2
\end{array} & \begin{array}{l}
\text { Project } \\
\# 3
\end{array} & \begin{array}{l}
\text { Project } \\
\# 4
\end{array} & \begin{array}{l}
\text { Project } \\
\# 5
\end{array} & \begin{array}{l}
\text { Project } \\
\# 6
\end{array} & \begin{array}{l}
\text { Project } \\
\# 7
\end{array} & \begin{array}{l}
\text { Project } \\
\# 8
\end{array} \\
\hline \text { Benefit } & 556 & 293 & 331 & 672 & 445 & 512 & 384 & 392 \\
\hline \text { Cost } & 348 & 189 & 294 & 460 & 327 & 392 & 282 & 296 \\
\hline \text { Duration } & 8 & 5 & 6 & 8 & 6 & 7 & 6 & 4 \\
\hline
\end{array}
$$
The company has decided to distribute the budget equally throughout the time horizon starting with the second month. Thus, an amount of $\bar{C}$=$$\$ 80,000$$ is allocated to each month $\mathrm{j}=2, \ldots, 24$. The cost incurred for each project is also equally distributed over the duration of the project. No cost is incurred during the set-up period.
Determine the project portfolio and the schedule of the selected projects for the problem stated above.