Answer to Question #257381 in Operations Research for elakiya

Question

An investor wants to identify how much to invest in two funds, one equity and one debt. Total amount available is Rs. 5, 00, 000. Not more than Rs. 3, 00, 000 should be invested in a single fund. Returns expected are 30% in equity and 8% in debt. Minimum return on total investment should be 15%. Formulate as LPP.

Answer 1

Let the amount to be invested in equity and debt fund be E and D respectively. These are the decision variables of the problem. We need to get maximum returns from the investment. As returns expected are 30% in equity and 8% in debt, the objective function which needs to be maximized is Z = 0.30E+0.08D

We now define the constraints of the problem. As the total amount available is 500000, the first constraint on total amount to be invested is as below

D+E$\leq 500000$

As we should not invest more than Rs. 300000 in a single fund we have the below constraint.

E$\leq 300000$

D $\leq 300000$

The third constraint is on the minimum returns required. We know that returns expected are 30% in equity and 8% in debt. Hence the total returns are 0.30E+0.08D. The total amount invested would be E+D. Hence the returns on total investment would be $\frac{(0.30E+0.08D)}{(E+D)}$ .We have the below constraint as minimum returns should be 15%

$\frac{0.30E+0.08D}{E+D} \geq 0.15$

${0.30E+0.08D} \geq 0.15(E+D)$

$0.30E+0.08D-0.15E-0.15D \geq 0$

$0.15E-0.07D\geq 0$

Hence the third constraint is $0.15E-0.07D \geq 0$

Also, we cannot invest negative amounts in any fund. Hence we have the non-negativity constraints as below

$E \geq 0, D \geq 0$

Thus the complete LPP formulation is as below

Maximize Z = 0.30E+0.08D subject to below constraints

$D+E\leq 500000$

$E \leq 300000$

$D \leq 300000$

$0.15E-0.07D \geq 0$

$E \geq 0, D \geq 0$