$3\mathrm{.}19.*$ Al Ferris has $$60,000$ that he wishes to invest now in order to use the accumulation for purchasing a retirement annuity in five years. After consulting with his financial advisor, he has been offered four types of fixed$-$income investments, which we will label as investments $A,B,C,$ and $D.$

Investments A and $B$ are available at the beginning of each of the next five 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)$ two 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 $$1\mathrm{.}90$ at the end of year $5.$ Each dollar invested in $D$ at the beginning of year $5$ returns $$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$.$ For this problem, all its functional constraints can be expressed as fixed$-$requirement constraints. To do this, let $A_v,B_v,C_D$ and $D_t,$ be the amounts invested in investments $A,B,C,$ and $D,$ respectively, at the beginning of year $t$ for each $t$ where the investment is available and will mature by the end of year $5.$ Also let $R_t$ 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