Varma Investment Services must develop an investment portfolio for a new client. As an initial investment strategy, the new client would like to restrict the portfolio to a mix of two stocks.

The client wants to invest $$44,000$ and established the following two investment goals.

Priority Level $1$ Goal

Goal $1$: Obtain an annual return of at least $9\%.$

Priority Level $2$ Goal

Goal $2$: Limit the investment in Key Oil, the riskier investment, to no more than $70\%$ of the total investment.

$($a$)$ Formulate a goal programming model for the Varma Investment problem. $($Let $x_i$ be the number of shares of stock i purchased, $d_{}$ be the deviation variable which exceeds the value of goal $i,d_{ni}$

be the deviation variable which is less than the value of goal $i,$ for $i=1,2.)$

Min $P_1()+P_2()$

s$.$t$.$

Funds Available $50x_1+100x_{2}44,000$

$P_1$ Goal $3x_1+10x_2-d_{p1}+d_{n1}=3960$

$P_2$ Goal $x_2-d_{p2}+d_{n2}=308$

$x_i,d_{n{i}^{\text{'}}}{d}_{}0$ for $i=1,2$

$($b$)$ Use the graphical goal programming procedure to obtain a solution.

$(x_1,x_2)=?$