Let's analyze the problem step by step. --- # **Given Data** ### Objective: Maximize profit: \[ \text{Max } Z = 40x_1 + 70x_2 \] ### Constraints: 1. Capacity constraint: \[ x_1 + x_2 \leq 100 \] 2. Budget constraint: \[ 10x_1 + 50x_2 \leq 400 \] 3. Non-negativity: \[ x_1, x_2 \geq \] ### Known optimal solutions: - Primal optimal solution: \(\mathbf{x} = (25, 75)\) - Dual variables: \(\mathbf{y} = (32.5, .75)\) --- # **Part (a): Borrowing to Increase Budget** ### Question: Should you borrow \$100 to increase your budget to \$410, paying back \$180? ### Step-by-step analysis: - **Current budget constraint:** \[ 10x_1 + 50x_2 \leq 400 \] - **Increased budget:** \[ 10x_1 + 50x_2 \leq 410 \] - **Cost of borrowing:** Borrow \$100, pay back \$180, net cost: \[ \$180 - \$100 = \$80 \] - **Impact on profit:** - Dual price (shadow price) of the budget constraint is the dual variable \( y_2 = .75 \). - **This means** for each additional dollar of budget, the profit increases by approximately \$.75, assuming the solution remains feasible. - **Additional budget:** \$100 - **Increase in profit (\( \Delta Z \)):** \[ \Delta Z = y_2 \times \text{additional budget} = .75 \times 100 = \$75 \] - **Net gain after repayment:** Since the profit increases by \$75, but you pay \$180 in total, the net change is: \[ \text{Net gain} = \$75 - \$180 = -\$105 \] ### **Conclusion for (a):** - The net profit decreases by \$105 after accounting for repayment. - **Thus, you should *not* borrow \$100 to increase the budget** because it results in a net loss. --- # **Part (b): Using \$100 to Increase Capacity** ### Question: Should you take \$100 from your current budget to increase capacity from 100 to 110 units? ### Step-by-step analysis: - **Current capacity constraint:** \[ x_1 + x_2 \leq 100 \] - **Proposed new capacity:** \[ x_1 + x_2 \leq 110 \] - **Cost of reallocating \$100:** - You are reducing your budget by \$100, which means you lose the ability to spend \$100 on production. - **Impact on profit:** - The dual variable \( y_1 = 32.5 \) (capacity constraint) indicates the marginal profit of increasing capacity by one unit. - **Additional capacity:** 10 units (from 100 to 110) - **Potential profit increase:** \[ \Delta Z = y_1 \times \text{additional units} = 32.5 \times 10 = \$325 \] - **Cost of reducing budget:** - Removing \$100 from the budget reduces potential profit by approximately \( y_2 \times 100 \) (though the exact impact depends on the new optimal solution). But since the original dual variable for capacity is high, increasing capacity seems beneficial. - **Trade-off:** - You gain an approximate profit increase of \$325. - You lose \$100 from your budget, which correspondingly reduces your ability to produce, but since the dual price \( y_2 \) is only .75, the profit loss from budget reduction is small. ### **Conclusion for (b):** - Because the potential profit gain (\$325) exceeds the loss from reducing \$100 of budget, **it is advantageous to reallocate \$100 from current budget to increase capacity**. --- # **Final Summary** | Part | Decision | Explanation | |---------|------------|--------------| | **(a)** | **No** | Borrowing \$100 to increase budget results in a net loss of \$105 after repayment, so it is not profitable. | | **(b)** | **Yes** | Reallocating \$100 from current budget to increase capacity yields a significant profit increase (approx. \$325), making it beneficial. | --- Let me know if you want the detailed calculations or further clarification!

# Summary of Each Step --- ## **Given Data** The objective is to maximize profit with constraints on capacity and budget. The primal optimal solution is \( (25, 75) \) and dual variables are \( (32.5, 0.75) \). --- ## **Part (a): Borrowing to Increase Budget** Borrowing \$100 to increase the budget results in a profit increase of \$75 but incurs a repayment cost of \$180. Thus, it leads to a net loss of \$105, making it unprofitable to borrow. --- ## **Part (b): Using \$100 to Increase Capacity** Reallocating \$100 from the budget to increase capacity yields an estimated profit increase of \$325. Since this gain far exceeds the loss from reducing the budget, it is advisable to make the capacity increase. --- ## **Final Summary** In part (a), borrowing is not recommended due to a net loss, while in part (b), reallocating funds for capacity increase is beneficial due to a significant profit gain. The analysis indicates a strategic decision-making approach to optimize production profits.