$3\mathrm{.}12-$ Maureen Laird is the chief financial officer for the Alva Electric Co$.,$ a major public utility in the Midwest. The company has scheduled the construction of new hydroelectric plants $5,10,$ and $20$ years from now to meet the needs of the growing population in the region served by the company. To cover the construction costs, Maureen needs to invest some of the company's money now to meet these future cash flow needs. Maureen may purchase only three kinds of financial assets, each of which costs $$1$ Million per unit. Fractional units may be purchased. The assets produce income $5,10,$ and $20$ years from now, and that income is needed to cover minimum cash flow requirements in those year, as shown in the following table.

$\backslash$table$[[$Year$,$Income per Unit od Asset,Minimum Cash,$],[$Asset $1,$Asset $2,$Flow Required$],[5,\backslash$table$[[$$$8],[$million$]],\backslash$table$[[$$$4],[$million$]],\backslash$table$[[$$$2],[$million$]],$$$1\mathrm{.}6$ billion$],[10,\backslash$table$[[$$$2],[$million$]],\backslash$table$[[$$$2],[$million$]],\backslash$table$[[$$$4],[$million$]],$$$400$ million$],[20,0,\backslash$table$[[$$$6],[$million$]],\backslash$table$[[$$$8],[$million$]],$$$1\mathrm{.}2$ billion$]]$

Maureen wishes to determine the mix of investments in these assets that will cover cash flow requirements while minimizing the total amount invested.

a $-$ Formulate a linear programming model for this problem on a spreadsheet.

b $-$ Use the spreadsheet to check the possibility of purchasing $100$ units of asset $1,100$ units of asset $2,$ and $200$ units of asset $3.$ How much cash flow would this mix of investments generate $5,10,$ and $20$ years from now? What would be the total amount invested?

c $-$ Take a few minutes to use a trial$-$and$-$error approach with the spreadsheet to develop your best guess for the optimal solution. What is the total amount invested for your solution?

d $-$ Use Solver to find an optimal solution.

e $-$ Summarize the model in algebraic form.