QUESTION $4$

$($a$)$ Write a GNU Octave code to determine the maximum value of $L,$ as well as the point $(x_1$;$x_2$;$x_3$;$x_4)$ at which the maximum occurs, of

$L=7x_1+6x_2+5x_3+4x_4,$

subject to the constraints

$x_1+x_2+x_3+x_4,,450000,$

$x_1,25000,$

$x_2,10000,$

$x_3,30000,$

$x_2+x_4,,70000,$

$x_1+x_2,,34000,$

$x_1,x_2,x_3,x_4,0.,$

$($b$)$ Suppose that a portfolio fund manager has investment portfolio with a total of $200$ Bonds, $120$ CFDs$,$ and $150$ Commodities. He is required to structure two types of investment baskets. Investment basket A contains $20$ Bonds, $10$ CFDs and $10$ Commodities. Investment basket B contains $10$ Bonds, $20$ CFDs and $30$ Commodities. If the profit $($return on investment$)$ of basket A is $$5000$ and that of basket B is $$12000,$ how many of each basket should he avail to investors in order to maximize the profit $($returns$)?$ To solve this maximization problem write a GNU Ocatve code that computes maximum number of per basket to be used in order to yield optimal returns.

$[25]$