# Linear Programming: Three variable graphical solution

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:$$X_1 \ge 625 \times 10^4X_1 \le 150 \times 10^5X_3 \ge \frac{25}{3} \times 10^6$$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" ## 1 Answer 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. • Hm, then we need another constraint: X_1 + X_2 + X_3 <= 25,000,000\$$ – Quixotic Sep 16 '13 at 14:24
• @Quixotic: Yes of course, I'll edit! – Mårten W Sep 16 '13 at 14:24