Let's solve the problem step by step. --- ### **Given Data** #### **Products** - Product M: - Selling Price = $190/unit - Max Demand = 100 units/week - Product N: - Selling Price = $200/unit - Max Demand = 80 units/week #### **Raw Materials (RM) Costs** - Product M: - RM-1 = $60/unit - RM-2 = $55/unit - Product N: - RM-1 = $60/unit - RM-3 = $40/unit #### **Machine Times per Unit** | Machine | Product M | Product N | |---------|-----------|-----------| | A | 15 min | 15 min | | B | 20 min | 5 min | | C | --- | 5 min | #### **Resources** - Each machine available: 2,400 min/week - Operating expenses: $12,000/week (fixed, not included in RM costs) - Ignore setup and transfer times --- ## **a. Which machine is the constraint?** Let's calculate the total machine time required for each machine if we produce the maximum demand for both products. #### **Machine A** - Product M: 15 min/unit × 100 units = 1,500 min - Product N: 15 min/unit × 80 units = 1,200 min - **Total: 1,500 + 1,200 = 2,700 min** (Exceeds 2,400 min available) #### **Machine B** - Product M: 20 min/unit × 100 units = 2,000 min - Product N: 5 min/unit × 80 units = 400 min - **Total: 2,000 + 400 = 2,400 min** (Exactly at capacity) #### **Machine C** - Product M: 0 min - Product N: 5 min/unit × 80 units = 400 min - **Total: 400 min** (Much less than 2,400 min) **Conclusion:** - **Machine A requires 2,700 min** (over capacity, so this is the constraint). - Machine B requires 2,400 min (at capacity, but not over). - Machine C requires only 400 min. > **Machine A is the constraint.** --- ## **b. Which product mix provides the highest gross profit?** ### **Step 1: Calculate Unit Contribution Margin (Gross Profit per Unit)** #### **Product M** - Revenue: $190 - RM-1: $60 - RM-2: $55 - **Total RM cost:** $115 - **Gross Profit per unit:** $190 - $115 = **$75** #### **Product N** - Revenue: $200 - RM-1: $60 - RM-3: $40 - **Total RM cost:** $100 - **Gross Profit per unit:** $200 - $100 = **$100** --- ### **Step 2: Gross Profit per minute ON THE CONSTRAINT (Machine A)** Both products use 15 min per unit on Machine A. - Product M: $75 / 15 min = **$5.00 per min** - Product N: $100 / 15 min = **$6.67 per min** **Product N generates more gross profit per minute on the constraint.** --- ### **Step 3: Find the Best Product Mix** - **Maximize Product N:** - Max demand = 80 units - Minutes required on Machine A = 80 × 15 = 1,200 min - **Minutes left for Product M:** - Machine A capacity = 2,400 min - Used for N = 1,200 min - Remaining = 1,200 min - **Product M units possible:** - 1,200 min ÷ 15 min/unit = 80 units - **But Product M max demand is 100 units, so 80 units is ok.** **Optimal Product Mix:** - **Product N:** 80 units/week (full demand) - **Product M:** 80 units/week (limited by machine A) --- ## **c. What is the maximum weekly net profit using the above mix?** ### **Step 1: Calculate Gross Profit** - **Product N:** 80 units × $100 = $8,000 - **Product M:** 80 units × $75 = $6,000 - **Total Gross Profit:** $8,000 + $6,000 = $14,000 ### **Step 2: Net Profit (after operating expenses)** - Operating Expenses: $12,000/week - **Net Profit = Gross Profit - Operating Expenses** - **Net Profit = $14,000 - $12,000 = $2,000** --- ## **Final Summary Table** | Question | Answer | |----------|--------| | a. Which machine is the constraint? | **Machine A** | | b. Which product mix provides the highest gross profit? | **80 units of M, 80 units of N per week** | | c. What is the maximum weekly net profit using this mix? | **$2,000 per week** | --- ## **Final Answers** ### a. Which machine is the constraint in this plant? > **Machine A** ### b. Which product mix provides the highest gross profit? > **Product Mix:** > Product M: **80 units** > Product N: **80 units** ### c. What is the maximum weekly net profit this plant can earn using the product mix from Part b? > **Weekly net profit:** **$2,000**

Let's analyze the problem step by step carefully to identify the correct answers. --- ### **Given Data Recap** **Products:** - **Product M:** - Selling Price = $190/unit - Demand = 100 units/week - **Product N:** - Selling Price = $200/unit - Demand = 80 units/week **Raw Materials and Costs:** - **Product M:** - RM-1 = $60/unit - RM-2 = $55/unit - **Product N:** - RM-1 = $60/unit - RM-3 = $40/unit **Machine Times per Unit:** | Machine | Product M | Product N | |---------|--------------|--------------| | A | 15 min | 15 min | | B | 20 min | 5 min | | C | --- | 5 min | **Resources and Costs:** - Each machine available: 2,400 min/week - Operating expenses: $12,000/week (fixed, excludes raw materials) - No setup/transfer times --- ## **a. Which machine is the constraint?** ### Step 1: Calculate total time required if producing maximum demand for both products: - **Machine A:** - Product M: 15 min × 100 units = 1,500 min - Product N: 15 min × 80 units = 1,200 min - **Total:** 1,500 + 1,200 = **2,700 min** (> 2,400 min capacity) - **Machine B:** - Product M: 20 min × 100 units = 2,000 min - Product N: 5 min × 80 units = 400 min - **Total:** 2,000 + 400 = **2,400 min** (equals capacity) - **Machine C:** - Product N only: 5 min × 80 units = 400 min ### **Conclusion:** - **Machine A** requires **2,700 min**, which exceeds capacity → **Machine A is the bottleneck.** --- ## **b. Which product mix provides the highest gross profit?** ### Step 1: Calculate **gross profit per unit** (ignoring operating costs): - **Product M:** - Revenue = $190 - Raw material costs = $60 + $55 = $115 - **Gross profit per unit:** $190 - $115 = **$75** - **Product N:** - Revenue = $200 - Raw material costs = $60 + $40 = $100 - **Gross profit per unit:** $200 - $100 = **$100** ### Step 2: Calculate **gross profit per minute** on the **constraint machine (Machine A)**: - Both products take 15 min per unit on Machine A. | Product | Gross profit per unit | Profit per minute (on Machine A) | |---------|-----------------------|-----------------------------------| | M | $75 | $75 / 15 min = **$5.00/min** | | N | $100 | $100 / 15 min = **$6.67/min** | **Product N** yields a higher gross profit per minute on the constraint. ### Step 3: Maximize profit on the constraint (Machine A): - To get maximum profit, prioritize **Product N** (profit per minute = $6.67). - **Maximum units of N on Machine A:** - Capacity = 2,400 min/week - Units = 2,400 min / 15 min per unit = **160 units** - But demand for Product N = 80 units (max demand), so produce **80 units** of N. - **Remaining time on Machine A:** - Used for N: 80 × 15 = 1,200 min - Remaining capacity: 2,400 - 1,200 = 1,200 min - **Produce Product M with remaining time:** - Units of M = 1,200 min / 15 min per unit = **80 units** - **Check demands:** - Demand for M: 100 units, but produce only 80 units (limited by capacity) - Demand for N: 80 units, fully met ### **Optimal product mix:** - **Product N:** 80 units - **Product M:** 80 units --- ## **c. What is the maximum weekly net profit?** ### Step 1: Calculate gross profit for this mix: - **Product N:** 80 units × $100 = **$8,000** - **Product M:** 80 units × $75 = **$6,000** - **Total gross profit:** **$8,000 + $6,000 = $14,000** ### Step 2: Subtract operating expenses: - Operating expenses = $12,000 - **Net profit = $14,000 - $12,000 = $2,000** --- ## **Final Summary** | Question | Answer | |----------|--------| | a. Which machine is the constraint? | **Machine A** | | b. Which product mix provides the highest gross profit? | **80 units of N, 80 units of M** | | c. What is the maximum weekly net profit? | **$2,000** | --- ## **Final statements:** **a.** Machine A is the constraint because producing the maximum demand for both products exceeds its capacity. **b.** To maximize gross profit, produce **80 units of Product N** (highest profit per minute) and **80 units of Product M** within the constraint. **c.** The maximum weekly net profit achievable with this mix is **$2,000**. --- **Note:** If demand or capacity were different, the optimal mix might change, but based on the analysis, this is the best solution.