A company is considering investing in four bonds; $$1$ million is available for investment. The expected annual return, the worst$-$case annual return on each bond, and the duration of each bond are given below:

$($The duration of a bond is a measure of the bond's sensitivity to interest rates.$)$ The company wants to maximize the expected return from its bond investments, subject to three constraints:

$*$ The worst$-$case return of the bond portfolio must be at least $9\%.$

$*$ The average duration of the portfolio must be at most $7.$

$*$ Because of diversification requirements, at most $35\%$ of the total amount invested can be invested in a single bond.

Determine how the company can maximize the expected return on its investment.

a$)$ What the the expected return of the optimized portfolio? $($answer in percent to three decimals$)$

$\%$

b$)$ What is the total expected return? $($no commas or $$-$sign$)$

c$)$ How much is invested in the lowest $($expected$)$ yielding bond? $($no commas or $$-$sign$)$

d$)$ Which, if any, bond$($s$)$ received no investment? $($answer with number or zero$)$