Dan Boyd is a financial planner trying to determine how to invest $$100,000$ for one of his clients. The cash flows for the five investments under consideration are summarized in the following table:

$\backslash$table$[[,\backslash$table$[[$Summary of Cash In$-$Flows and Out$-$Flows$],[($at Beginning of Year$)]]],[,$A$,$B$,$C$,$D$,$E$],[$Year $1,-1\mathrm{.}00,0\mathrm{.}00,-1\mathrm{.}00,0\mathrm{.}00,-1\mathrm{.}00],[$Year $2,+0\mathrm{.}45,-1\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00],[$Year $3,+1\mathrm{.}05,0\mathrm{.}00,0\mathrm{.}00,-1\mathrm{.}00,1\mathrm{.}25],[$Year $4,0\mathrm{.}00,+1\mathrm{.}30,+1\mathrm{.}65,+1\mathrm{.}30,0\mathrm{.}00]]$

For example, if Dan invests $$1$ in investment $A$ at the beginning of year $1,$ he will receive $$0\mathrm{.}45$ at the beginning of year $2$ and another $$1\mathrm{.}05$ at the beginning of year $3.$ Alternatively, he can invest $$1$ in investment $B$ at the beginning of year $2$ and receive $$1\mathrm{.}30$ at the beginning of year $4.$ Entries of $"0\mathrm{.}00"$ in the preceding table indicate times when no cash in$-$flows or out$-$flows can occur. The minimum required investment for each of the possible investments is $$50,000.$ Also, at the beginning of each year, Dan

may also place any or all of the available money in a money market account that is expected to yield $5\%$ per year. How should Dan plan his investments if he wants to maximize the amount of money available to his client at the end of year $4?$

a$.$ Formulate an ILP model for this problem.

b$.$ Create a spreadsheet model for this problem and solve it using Solver.

c$.$ What is the optimal solution?