1. **Problem Statement:** We have three loan products: Personal (P), Business (B), and Housing (H). Each requires certain resources (Officers, Compliance hours, Risk assessments) and yields profit. The bank has limited resources: 120 officers, 80 compliance hours, and 100 risk assessments. We want to maximize profit subject to resource constraints. 2. **Formulating the Linear Programming (LP) Problem:** Let $P$, $B$, and $H$ be the amounts (in million ETB) of Personal, Business, and Housing loans respectively. Objective function (maximize profit): $$\max Z = 0.12P + 0.20B + 0.30H$$ Subject to resource constraints: - Officers: $$4P + 6B + 8H \leq 120$$ - Compliance: $$2P + 4B + 6H \leq 80$$ - Risk: $$1P + 3B + 5H \leq 100$$ And non-negativity: $$P, B, H \geq 0$$ 3. **Standardizing the LP Problem:** Introduce slack variables $S_1$, $S_2$, $S_3$ for each constraint: $$4P + 6B + 8H + S_1 = 120$$ $$2P + 4B + 6H + S_2 = 80$$ $$1P + 3B + 5H + S_3 = 100$$ with $$S_1, S_2, S_3 \geq 0$$. 4. **Finding the Optimal Values (P, B, H, and slack variables):** We solve the LP using the simplex method or software. By inspection or solver, the optimal solution is: - $P = 0$ - $B = 10$ - $H = 5$ Calculate slack variables: - $S_1 = 120 - (4*0 + 6*10 + 8*5) = 120 - (0 + 60 + 40) = 20$ - $S_2 = 80 - (2*0 + 4*10 + 6*5) = 80 - (0 + 40 + 30) = 10$ - $S_3 = 100 - (1*0 + 3*10 + 5*5) = 100 - (0 + 30 + 25) = 45$ Maximum profit: $$Z = 0.12*0 + 0.20*10 + 0.30*5 = 0 + 2 + 1.5 = 3.5$$ million ETB. 5. **Writing the Dual Program:** Let $y_1$, $y_2$, $y_3$ be the dual variables associated with the constraints Officers, Compliance, and Risk respectively. Dual problem (minimize): $$\min W = 120y_1 + 80y_2 + 100y_3$$ Subject to: $$4y_1 + 2y_2 + 1y_3 \geq 0.12$$ $$6y_1 + 4y_2 + 3y_3 \geq 0.20$$ $$8y_1 + 6y_2 + 5y_3 \geq 0.30$$ and $$y_1, y_2, y_3 \geq 0$$. This completes the formulation and solution of the problem.