Let's break down the problem and **solve the Integer Linear Programming (ILP) step-by-step**: --- ## **Step 1: Decision Variables** Let \( y_i = 1 \) if product \( i \) is selected, \( 0 \) otherwise (for \( i = 1 \) to \( 7 \)). --- ## **Step 2: Objective Function** Maximize Total Profit: \[ \text{Maximize } Z = 120y_1 + 100y_2 + 90y_3 + 85y_4 + 125y_5 + 105y_6 + 140y_7 \] --- ## **Step 3: Constraints** ### **Budget** \[ 750y_1 + 650y_2 + 500y_3 + 450y_4 + 800y_5 + 700y_6 + 1000y_7 \leq 5000 \] ### **Labor Hours** \[ 20y_1 + 18y_2 + 15y_3 + 12y_4 + 22y_5 + 20y_6 + 30y_7 \leq 150 \] ### **Bottle Supply** \[ 40y_1 + 35y_2 + 30y_3 + 25y_4 + 40y_5 + 35y_6 + 45y_7 \leq 250 \] ### **Product Selection Limit** \[ y_1 + y_2 + y_3 + y_4 + y_5 + y_6 + y_7 \leq 4 \] ### **Logical Constraints** 1. **Avoid overlap (Argan vs Coconut + Rosemary):** \( y_1 + y_2 \leq 1 \) 2. **At least one floral scent (Rose, Rosewater):** \( y_5 + y_6 \geq 1 \) 3. **Luxury Gold Blend only if Argan product selected:** \( y_7 \leq y_1 + y_5 \) --- ## **Step 4: Solving the ILP** Let's **analyze combinations** to maximize profit while satisfying all constraints. ### **Step 4.1: Which products are strongest?** - **Highest profit per product:** Y7 (140), Y5 (125), Y6 (105), Y1 (120) - **But Y7 needs Y1 or Y5** #### Let's try to select Y7, Y5, Y6, plus one more ### **Trial 1: Y5, Y6, Y7, Y1** - \( y_1 = 1, y_5 = 1, y_6 = 1, y_7 = 1 \) (all others = 0) Let's check constraints: - **Budget:** \( 750 + 800 + 700 + 1000 = 3250 \leq 5000 \) ✔️ - **Labor:** \( 20 + 22 + 20 + 30 = 92 \leq 150 \) ✔️ - **Bottles:** \( 40 + 40 + 35 + 45 = 160 \leq 250 \) ✔️ - **Product limit:** \( 4 \leq 4 \) ✔️ - **No overlap (Y1 + Y2):** \( 1+0=1 \leq 1 \) ✔️ - **At least one floral (Y5 + Y6):** \( 1+1=2 \geq 1 \) ✔️ - **Luxury only if Argan base:** \( y_7 \leq y_1 + y_5 \Rightarrow 1 \leq 2 \) ✔️ **Profit:** \( 120 + 125 + 105 + 140 = 490 \) --- ### **Can we get more profit by swapping Y1 for another?** #### Try: Y2, Y5, Y6, Y7 (swap Y1 for Y2) - This would violate the luxury blend constraint since Y2 isn't argan-based. - \( y_7 \leq y_1 + y_5 \) would become \( 1 \leq 0+1 \) (OK), but is Coconut+Rosemary better? **Profit:** \( 100 + 125 + 105 + 140 = 470 \) (lower than 490) --- ### **Try: Y1, Y5, Y7, Y4 (swap Y6 with Y4)** - **Profit:** \( 120 + 125 + 140 + 85 = 470 \) (lower than 490) - **Floral constraint:** Still okay (Y5 is floral) --- ### **Try: Y1, Y5, Y7, Y3** - **Profit:** \( 120 + 125 + 140 + 90 = 475 \) (lower than 490) --- ### **Try: Y1, Y6, Y7, Y3** - **Profit:** \( 120 + 105 + 140 + 90 = 455 \) (lower than 490) --- ### **Try: Y1, Y5, Y6, Y2 (replace Y7 with Y2)** - **Profit:** \( 120 + 125 + 105 + 100 = 450 \) (lower than 490) --- ## **Step 5: Double-check constraints for best combo (Y1, Y5, Y6, Y7)** - All constraints satisfied. - **Max Profit = 490** --- ## **Step 6: Solution** ### **Final Answer** **Select the following 4 products for launch:** - **Y1: Argan+Rosemary** - **Y5: Argan+Rose** - **Y6: Rosewater Hydration Mix** - **Y7: Luxury Gold Blend** ### **Maximum Total Profit = \$490** **Product Selection Vector:** \( (y_1, y_2, y_3, y_4, y_5, y_6, y_7) = (1, 0, 0, 0, 1, 1, 1) \) --- ### **Reasoning:** - This combo maximizes profit while satisfying all numerical and logical constraints. - All constraints (budget, labor, bottles, overlap, floral, luxury) are respected. --- **If you need the step-by-step logic for each constraint, just ask!**