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A small bank offers three type of loans: housing loans at $8.50$% interest, education loans at $13.75$% interest rates, and loans to senior citizens at $12.25$% interest. Further, it needs to adhere to certain policy restrictions. The restrictions require the bank to ensure that Condition 1: housing loans make up between $25$% and $60$% of the total loan amount disbursed;and Condition 2: the amount of loans disbursed to senior citizens should be least one third of the total amount disbursed as loans. In a particular year, its lending capacity is $25,000,000\$$. The bank would like to disburse loans so as to maximize its earnings from the interest paid. Solve the problem graphically by taking two at a time.

From the problem, I derive the following LP model:

Decision variables:

$X_1:$ Amount disbursed as housing loan.
$X_2:$ Amount disbursed as education loan.
$X_3:$ Amount disbursed as loans to senior citizens.

Maximize $$0.085X_1 + 0.1375X_2 + 0.1225X_3$$

Subject to constraints:

$$X_1 \ge 625 \times 10^4$$ $$X_1 \le 150 \times 10^5$$ $$X_3 \ge \frac{25}{3} \times 10^6$$ where $X_1,X_2,X_3 \ge 0$

My question is: Is this formulation correct? and how to solve this three variable equations using graphical method. Precisely I don't understand "Solve the problem graphically by taking two at a time"

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It looks sensible, except I think your constraints should be $$ \begin{aligned} X_1&\geq0.25(X_1+X_2+X_3) \\ X_1&\leq0.6(X_1+X_2+X_3) \\ X_3&\geq\frac{1}{3}(X_1+X_2+X_3) \\ X_1+X_2+X_3&\leq25000000 \\ X_1,X_2,X_3&\geq0 \end{aligned} $$ Of course, at the optimal solution, the bank will lend all it can, but that is for the optimisation to take care of.

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  • $\begingroup$ Hm, then we need another constraint: $X_1 + X_2 + X_3 <= 25,000,000\$$ $\endgroup$ – Quixotic Sep 16 '13 at 14:24
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    $\begingroup$ @Quixotic: Yes of course, I'll edit! $\endgroup$ – Mårten W Sep 16 '13 at 14:24

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