Al Ferris has $$60,000$ that he wishes to invest now in or$-$

der to use the accumulation for purchasing a retirement annuity in

$5$ years. After consulting with his financial adviser, he has been

offered four types of fixed$-$income investments, which we will la$-$

bel as investments A$,$ B$,$ C$,$ D$.$

Investments A and B are available at the beginning of each

of the next $5$ years $($call them years $1$ to $5).$ Each dollar invested

in A at the beginning of a year returns $$1\mathrm{.}40($a profit of $$0\mathrm{.}40)$

$2$ years later $($in time for immediate reinvestment$).$ Each dollar

invested in B at the beginning of a year returns $$1\mathrm{.}70$ three years

later.

Investments C and D will each be available at one time in the

future. Each dollar invested in C at the beginning of year $2$ returns

$$1\mathrm{.}90$ at the end of year $5.$ Each dollar invested in D at the begin$-$

ning of year $5$ returns $$1\mathrm{.}30$ at the end of year $5.$

Al wishes to know which investment plan maximizes the

amount of money that can be accumulated by the beginning of

year $6.$

$($a$)$ All the functional constraints for this problem can be expressed

as equality constraints. To do this, let At$,$ Bt$,$ Ct$,$ and Dt be the

amount invested in investment A$,$ B$,$ C$,$ and D$,$ respectively, at

the beginning of year t for each t where the investment is avail$-$

able and will mature by the end of year $5.$ Also let Rt be the

number of available dollars not invested at the beginning of

year t $($and so available for investment in a later year$).$ Thus,

the amount invested at the beginning of year t plus Rt must

equal the number of dollars available for investment at that

time. Write such an equation in terms of the relevant variables

above for the beginning of each of the $5$ years to obtain the five

functional constraints for this problem.

$($b$)$ Formulate a complete linear programming model for this

problem.

C $($c$)$ Solve this model by the simplex model.