A bond portfolio manager has $$100,000$ to allocate to two different bonds: one corporate bond, and one government bond.

The corporate bond has a yield of $4\%,$ a maturity of $3$ years and an A rating from a rating agency. For computational purposes, this A rating translates into a numerical rating of $2.$ By contrast, the government bond has a yield of $3\%,$ a maturity of $4$ years and rating of Aaa, which has a corresponding numerical rating of $1.($Lower numerical ratings correspond to higher quality bonds$).$

The portfolio manager would like to allocate her funds so that the average rating for the portfolio is no worse than Aa $($numerical equivalent $1\mathrm{.}5)$ and average maturity of the portfolio is at most $3\mathrm{.}6$ years. Any amount not invested in the two bonds will be kept in a cash account. The yield of this cash account is $0\mathrm{.}01\%.$ The cash account also does not contribute to the average rating or maturity computations $($in other words, we are assuming a quality rating of $0$ with $$perfect$$ quality and maturity of $0$ years for cash$).$

How should the manager allocate her funds between these two bonds to achieve her objective of maximizing the yield from this investment above the yield she would have by putting all her money in the cash account? $[$ie maximise: $($yield from bondS AND cash$)-($yield from cash only if she leaves all money in account$)]$

$($a$)$ Let variables x$1$ and x$2$ denote the allocation of funds to the corporate and government bond respectively $($in thousands of dollars$).$ Write down the portfolio manager$$s problem as a linear programming problem in terms of x$1,$x$2,$ in standard form.

$($b$)$ Introduce slack or surplus variables $($whatever you feel is appropriate$),$ and write down the LP problem. $($You are not required to solve it$).$