Let's solve the problem step by step. --- ## (a) **Formulate the Linear Programming Model** **Let:** - \( H \) = amount allocated to home loans - \( P \) = amount allocated to personal loans - \( A \) = amount allocated to automobile loans ### Objective Function Maximize total annual return: \[ \text{Maximize } Z = 0.04H + 0.10P + 0.07A \] ### Constraints 1. **At least 40% to home loans:** \[ H \geq 0.4 \times 1{,}000{,}000 = 400{,}000 \] 2. **Personal loans cannot exceed 60% of amount allocated to auto loans:** \[ P \leq 0.6A \] 3. **Total funds allocated:** \[ H + P + A = 1{,}000{,}000 \] 4. **Non-negativity:** \[ H \geq 0, \quad P \geq 0, \quad A \geq 0 \] --- ## (b) **How much should be allocated to each type of loan?** Let's solve the system: ### 1. Substitute \( P = 0.6A \) (since maximizing return, allocate as much as possible to higher return loans within constraint bounds). ### 2. \( H + P + A = 1,000,000 \Rightarrow H + 0.6A + A = 1,000,000 \Rightarrow H + 1.6A = 1,000,000 \) ### 3. \( H \geq 400,000 \) ### 4. \( H \geq 0, \quad A \geq 0 \) To maximize return, allocate minimum possible to lowest return (Home, 4%) and as much as possible to the highest return within constraints (Personal, 10%). Set \( H = 400,000 \): \[ 400,000 + 1.6A = 1,000,000 \implies 1.6A = 600,000 \implies A = 375,000 \] Then, \[ P = 0.6A = 0.6 \times 375,000 = 225,000 \] --- ### **Final Allocations:** | Loan Type | Allocation (\$) | |------------|-----------------| | Home | 400,000 | | Personal | 225,000 | | Automobile | 375,000 | --- ### **Total Annual Return:** \[ \text{Total return} = 0.04 \times 400,000 + 0.10 \times 225,000 + 0.07 \times 375,000 \] \[ = 16,000 + 22,500 + 26,250 = \boxed{64,750} \] --- ### **Annual Percentage Return:** \[ \text{Annual Percentage Return} = \frac{64,750}{1,000,000} \times 100 = \boxed{6.48\%} \] --- ### **Summary Table for (b):** | Loan Type | Allocation (\$) | |------------|-----------------| | Home | 400,000 | | Personal | 225,000 | | Automobile | 375,000 | - **Total annual return:** \$64,750 - **Annual percentage return:** 6.48% --- ## (c) **If the interest rate on home loans increases to 9%...** Home loans now return 9%, so the new objective function: \[ \text{Maximize } Z = 0.09H + 0.10P + 0.07A \] Now, home loans are almost as profitable as personal loans. To maximize return, allocate the minimum possible to the lowest return (now auto, 7%) and as much as possible to home and personal. But the constraint \( P \leq 0.6A \) still limits personal loans. Let’s try \( A = 0 \), \( P = 0 \), \( H = 1,000,000 \): Check constraints: - \( H \geq 400,000 \) ✔️ - \( H + P + A = 1,000,000 \) ✔️ So, all funds to home loans is feasible and gives: \[ \text{Total return} = 0.09 \times 1,000,000 = 90,000 \] \[ \text{Annual percentage return} = 9\% \] **So yes, the allocations would change: All funds would be allocated to home loans.** --- ## (d) **Suppose the total amount of new funds increases by \$10,000...** All allocations are proportional, so: - New total = \$1,010,000 - New allocations: - Home: 40% of \$1,010,000 = \$404,000 - Auto: \$375,000 / \$1,000,000 × \$1,010,000 = \$378,750 - Personal: \$225,000 / \$1,000,000 × \$1,010,000 = \$227,250 - New annual return = \$64,750 / \$1,000,000 × \$1,010,000 = \$65,397.50 - Increase in total return = \$65,397.50 - \$64,750 = **\$647.50** **Each additional \$1,000 added increases annual return by \$64.75.** --- ## (e) **If minimum home loan allocation is relaxed to 39%...** - Home: \( H = 0.39 \times 1,000,000 = 390,000 \) - \( H + 1.6A = 1,000,000 \implies 390,000 + 1.6A = 1,000,000 \implies 1.6A = 610,000 \implies A = 381,250 \) - \( P = 0.6 \times 381,250 = 228,750 \) Total return: \[ 0.04 \times 390,000 + 0.10 \times 228,750 + 0.07 \times 381,250 = 15,600 + 22,875 + 26,687.50 = 65,162.50 \] Annual percentage return: \[ \frac{65,162.50}{1,000,000} \times 100 = 6.52\% \] **So, relaxing the home loan minimum to 39% increases the annual return by \$412.50 and the percentage to 6.52%.** --- ## **Summary of Answers:** ### (b) Allocations: - Home: **\$400,000** - Personal: **\$225,000** - Automobile: **\$375,000** ### Total annual return: **\$64,750** ### Annual percentage return: **6.48%** --- Let me know if you need the answers entered in the boxes or want explanations for the other parts!