Q$3.(30$ pts$)($Capital Budgeting$)$ The U$.$S$.$ space agency, NASA, must deal constantly with such

decision problems in choosing how to divide its limited budgets among many competing missions

proposed. The table below shows a fictitious list of alternatives.

NASA needs to decide which of the $14$ indicated missions to include in program plans over $5$

stages of a $25-$year era. The overall objective is to maximize the total value of selected missions,

by considering the following constraints:

Budget constraints: limit the total funds or other resources consumed by selected projects,

investments, and so on$,$ in each time period not to exceed the amount available.

Mutually exclusive constraints: two or more proposed projects may be mutually exclusive.

For example, Possibilities $j=4$ and $5$ represent alternative timing for the same mission.

Technologies for missions $j=8$ and $11$ are incompatible. Numbers $j=9$ and $14$ involve two

different ways to accomplish the SETI program.

Dependency constraints: Another characteristic relationship between projects arises when

one project depends on another. We cannot choose such a dependent project unless we also

include the option on which it depends. The dependency relationship is shown in the last

column of the table.

a$.$ Formulate an integer problem for the capital budgeting problem by integrating the above

constraints. Define clearly all the notations used and explain the constraints.

b$.$ Solve the problem with Excel$-$Solver or AMPL optimization software, report the optimal cities

to locate DCs in and the optimal total annual cost

Note: you only need to solve it using one software of your choice, no need to solve it with both.

When submitting Excel file, please make sure your model setting is complete, solver setting

is included, and the final results are displayed.

When submitting AMPL files, please define everything clearly, including the mod, dat,

and run files.