$4.$ Dynamic Programming

$(15$ Marks$)$

Consider that we have to make an investment decision. Total amount of available money for investment is $\backslash (\backslash$$ $70,000\backslash )$ and there are $4$ different investment options. The associated return for each option is given by $\backslash (\backslash$$ $30,000\backslash )($Option $1),\backslash$$$40,000($Option $2),\backslash (\backslash$$ $50,000\backslash )($Option $3),$ and $\backslash$$$70,000($Option $4)$ respectively. However, the probability for obtaining the returns are as follows: $0\mathrm{.}9($Option $1),0\mathrm{.}8($Option $2),0\mathrm{.}7($Option $3),$ and $0\mathrm{.}5($Option $4).$ Each option also has a purchase amount associated with it given by $\backslash (\backslash$$ $20,000\backslash )($Option $1),\backslash$$$30,000($Option $2),\backslash$$$40,000($Option $3),$ and $\backslash$$$50,000($Option $4)$ respectively. Formulate the problem in the framework of Dynamic Programming and find the investment with maximum return. Multiple options can be purchased but one option can be purchased only once.