Let's break down your problem step by step: --- ### **Given Data** #### **Resources:** - **Machines:** A, B, C (1 each) - **Availability:** 2,400 min/week per machine #### **Products:** | Product | Selling Price | Max Demand/week | RM-1 | RM-2 | RM-3 | Machine A (min) | Machine B (min) | Machine C (min) | |---------|--------------|----------------|------|------|------|-----------------|-----------------|-----------------| | M | $190 | 100 units | $60 | - | - | 15 | 20 | 15 | | N | $200 | 80 units | - | $40 | $55 | 15 | 5 | 5 | #### **Operating Expenses:** $12,000/week (not including raw materials) --- ## **a. Which machine is the constraint?** Let's calculate total required minutes for each machine if we make the maximum demand of each product. ### **If we make 100 M and 80 N (max demand):** #### **Machine A** - M: 100 units × 15 min = **1,500 min** - N: 80 units × 15 min = **1,200 min** - **Total:** 2,700 min #### **Machine B** - M: 100 × 20 = **2,000 min** - N: 80 × 5 = **400 min** - **Total:** 2,400 min #### **Machine C** - M: 100 × 15 = **1,500 min** - N: 80 × 5 = **400 min** - **Total:** 1,900 min **Constraint is the machine required for the highest time:** - **Machine A:** Needs 2,700 min (exceeds available 2,400 min, so is the constraint) - **Machine B:** 2,400 min (at the limit) - **Machine C:** 1,900 min (has slack) **However, both A and B exceed or reach the limit. Let's check which is the tightest constraint.** But, if we try to make both products at max demand, Machine A needs 2,700 min, which is more than 2,400 min, so **Machine A is the constraint**. --- ## **b. Which product mix provides the highest gross profit?** ### **Gross profit per unit (ignore operating expense):** #### **Product M:** - Selling Price = $190 - Raw Material = $60 - **Gross Profit per unit = $190 - $60 = $130** #### **Product N:** - Selling Price = $200 - Raw Material = $40 + $55 = $95 - **Gross Profit per unit = $200 - $95 = $105** ### **BUT:** We must use the constraint (Machine A) to decide the best mix. #### **Minutes on A per unit:** - Both M and N need 15 min on A #### **Profit per minute of A:** - M: $130 / 15 min = **$8.67 per min** - N: $105 / 15 min = **$7.00 per min** **So, maximizing Product M uses the constraint most profitably.** But, demand for M is only 100 units (needs 1,500 min of A). **A's total available time:** 2,400 min/week - Use 1,500 min for 100 units of M - Remaining: 2,400 - 1,500 = **900 min** for N - N uses 15 min/unit on A - 900 min / 15 min/unit = **60 units of N** **So, product mix is:** - **100 units of Product M** - **60 units of Product N** --- ## **c. What is the maximum weekly net profit?** ### **Total Sales:** - M: 100 × $190 = $19,000 - N: 60 × $200 = $12,000 - **Total Sales: $31,000** ### **Raw Material Costs:** - M: 100 × $60 = $6,000 - N: 60 × ($40 + $55) = 60 × $95 = $5,700 - **Total Raw Materials = $6,000 + $5,700 = $11,700** ### **Gross Profit:** - $31,000 - $11,700 = **$19,300** ### **Subtract Operating Expenses:** - $19,300 - $12,000 = **$7,300** --- ## **Final Summary Table** | Question | Answer | |----------|--------| | a. Which machine is the constraint? | **Machine A** | | b. Which product mix maximizes gross profit? | **100 units of M, 60 units of N** | | c. What is the maximum net profit? | **$7,300 per week** | --- ### **Summary** - **Machine A** is the bottleneck/constraint. - **Best product mix:** 100 units of M and 60 units of N per week. - **Maximum weekly net profit:** **$7,300** If you need to see the calculations in table form or want the process laid out differently, please let me know!

Let's carefully analyze the problem step by step to answer each question thoroughly. --- ### **Given Data Overview** **Resources & Constraints:** - Machines A, B, C (each available 2,400 minutes/week) - Raw materials: - RM-1: $60/unit - RM-2: $40/unit - RM-3: $55/unit - Operating expense: $12,000/week - Demand: - Product M: 100 units/week - Product N: 80 units/week - Selling prices: - M: $190/unit - N: $200/unit **Process times per unit:** | Product | Machine A (min) | Machine B (min) | Machine C (min) | RM-1 | RM-2 | RM-3 | |---------|-----------------|-----------------|-----------------|-------|-------|-------| | M | 15 | 20 | 15 | $60 | - | - | | N | 15 | 5 | 5 | - | $40 | $55 | --- ### **a. Which machine is the constraint?** **Step 1:** Calculate total time needed if producing maximum demand: - **Product M:** 100 units - **Product N:** 80 units **Machine A:** - M: 100 × 15 = 1,500 min - N: 80 × 15 = 1,200 min - **Total A:** 1,500 + 1,200 = **2,700 min** **Machine B:** - M: 100 × 20 = 2,000 min - N: 80 × 5 = 400 min - **Total B:** 2,000 + 400 = **2,400 min** **Machine C:** - M: 100 × 15 = 1,500 min - N: 80 × 5 = 400 min - **Total C:** 1,500 + 400 = **1,900 min** **Available time per week:** 2,400 min **Analysis:** - Machine A needs 2,700 min → exceeds available time → **Constraint** - Machine B needs exactly 2,400 min → at limit - Machine C needs 1,900 min → within limit **Conclusion:** **Machine A** is the bottleneck (constraint). --- ### **b. Which product mix provides the highest gross profit?** **Step 1:** Calculate gross profit per unit (ignoring operating expenses): | Product | Selling Price | Raw Material Cost | Gross Profit per Unit | Profit per Min (on A) | |---------|--------------|------------------|------------------------|------------------------| | M | $190 | $60 | $130 | $130/15 = **$8.67** | | N | $200 | $40 + $55 = $95 | $105 | $105/15 = **$7.00** | **Step 2:** Use the constraint (Machine A's time) to maximize profit: - Max demand for M: 100 units (uses 1,500 min on A) - Remaining A time: 2,400 - 1,500 = 900 min for N - N units possible: 900 / 15 = 60 units **Step 3:** Product mix for maximum profit: - **100 units of M** - **60 units of N** --- ### **c. What is the maximum weekly net profit?** **Step 1:** Calculate total revenue: | Product | Units | Revenue | |---------|--------|------------------------| | M | 100 | 100 × $190 = **$19,000** | | N | 60 | 60 × $200 = **$12,000** | - **Total Revenue:** $19,000 + $12,000 = **$31,000** **Step 2:** Raw material costs: | Product | Units | RM-1 Cost | RM-2 Cost | RM-3 Cost | Total RM Cost | |---------|--------|------------|------------|------------|--------------| | M | 100 | 100 × $60 = $6,000 | - | - | $6,000 | | N | 60 | - | 60 × $40 = $2,400 | 60 × $55 = $3,300 | $2,400 + $3,300 = $5,700 | - **Total raw material cost:** $6,000 + $5,700 = **$11,700** **Step 3:** Gross profit: - Revenue - Raw material costs = $31,000 - $11,700 = **$19,300** **Step 4:** Subtract operating expenses: - Operating expenses: $12,000/week **Net profit:** - $19,300 - $12,000 = **$7,300** --- ### **Final Summary** | Question | Answer | |------------|---------| | **a. Which machine is the constraint?** | **Machine A** | | **b. Which product mix maximizes gross profit?** | **100 units of Product M and 60 units of Product N** | | **c. What is the maximum weekly net profit?** | **$7,300** | --- **In conclusion:** - The **constraint** is **Machine A**. - The **optimal product mix** to maximize profit is **100 units of M** and **60 units of N**. - The **maximum weekly net profit** achievable with this mix is **$7,300**. If you'd like further details or calculations, feel free to ask!

Certainly! Here's a clear and accurate solution to your problem: --- ### **Given Data Summary** | Aspect | Details | |---|---| | **Machines** | A, B, C (each with 2,400 min/week) | | **Raw Material Costs** | RM-1: $60/unit, RM-2: $40/unit, RM-3: $55/unit | | **Products** | M and N | | **Demand** | M: 100 units/week, N: 80 units/week | | **Prices** | M: $190/unit, N: $200/unit | | **Process times per unit** | | | Product | Machine A (min) | Machine B (min) | Machine C (min) | Raw Material Usage | |---|---|---|---|---| | M | 15 | 20 | 15 | RM-1: $60 | | N | 15 | 5 | 5 | RM-2: $40, RM-3: $55 | | **Operating Expenses** | $12,000/week | --- ### **a. Which machine is the constraint?** **Step 1:** Calculate total machine time needed for maximum demand: - **Product M:** 100 units - **Product N:** 80 units **Machine A:** - M: 100 × 15 = 1,500 min - N: 80 × 15 = 1,200 min - **Total:** 1,500 + 1,200 = **2,700 min** **Machine B:** - M: 100 × 20 = 2,000 min - N: 80 × 5 = 400 min - **Total:** 2,000 + 400 = **2,400 min** **Machine C:** - M: 100 × 15 = 1,500 min - N: 80 × 5 = 400 min - **Total:** 1,500 + 400 = **1,900 min** **Available time per week:** 2,400 min **Analysis:** - Machine A requires **2,700 min**, which **exceeds** the available 2,400 min → **Machine A is the constraint**. - Machine B is exactly at capacity if producing max demand. - Machine C has slack. **Answer:** **Machine A** is the constraint. --- ### **b. Which product mix provides the highest gross profit?** **Step 1:** Calculate profit per unit (excluding operating expenses): | Product | Selling Price | Raw Material Cost | Gross Profit per Unit | Profit per Min on A | |---|---|---|---|---| | M | $190 | $60 | $130 | $130 / 15 = **$8.67** | | N | $200 | $40 + $55 = $95 | $105 | $105 / 15 = **$7.00** | **Step 2:** Maximize profit within the constraint (Machine A): - To maximize profit, produce as many units of the higher profit/min product (Product M) as possible: - Max demand for M: 100 units (uses 1,500 min on A) - Remaining A time: 2,400 - 1,500 = 900 min - Units of N that can be produced: 900 / 15 = 60 units **Product mix:** - **100 units of M** - **60 units of N** --- ### **c. What is the maximum weekly net profit?** **Step 1:** Revenue: - M: 100 × $190 = **$19,000** - N: 60 × $200 = **$12,000** - **Total Revenue:** $19,000 + $12,000 = **$31,000** **Step 2:** Raw material costs: - M: 100 × $60 = $6,000 - N: 60 × ($40 + $55) = 60 × $95 = **$5,700** - **Total raw materials:** $6,000 + $5,700 = **$11,700** **Step 3:** Gross profit: - $31,000 - $11,700 = **$19,300** **Step 4:** Subtract operating expenses: - $19,300 - $12,000 = **$7,300** --- ### **Final Answers:** | Question | Answer | |---|---| | **a. Which machine is the constraint?** | **Machine A** | | **b. Which product mix provides the highest gross profit?** | **100 units of Product M and 60 units of Product N** | | **c. What is the maximum weekly net profit?** | **$7,300** | --- **Summary:** The bottleneck is Machine A. Producing 100 units of M and 60 units of N maximizes profit given the constraint. The maximum weekly net profit achievable is **$7,300**. If you'd like further details or clarification, feel free to ask!

Certainly! Here's a comprehensive and correct solution to your problem: --- ### **Given Data Summary** | Aspect | Details | |---|---| | **Machines** | A, B, C (each available 2,400 minutes/week) | | **Raw Material Costs** | RM-1: $60/unit, RM-2: $40/unit, RM-3: $55/unit | | **Products** | M and N | | **Demand** | M: 100 units/week, N: 80 units/week | | **Prices** | M: $190/unit, N: $200/unit | | **Process times per unit** | | | Product | Machine A (min/unit) | Machine B (min/unit) | Machine C (min/unit) | Raw Material Usage | |---|---|---|---|---| | M | 15 | 20 | 15 | RM-1: $60 | | N | 15 | 5 | 5 | RM-2: $40, RM-3: $55 | | **Operating Expenses** | $12,000/week | --- ### **a. Which machine is the constraint?** **Step 1:** Calculate total time required to meet maximum demand: - **Product M:** 100 units - **Product N:** 80 units **Machine A:** - M: 100 × 15 = 1,500 min - N: 80 × 15 = 1,200 min - **Total A:** 1,500 + 1,200 = **2,700 min** **Machine B:** - M: 100 × 20 = 2,000 min - N: 80 × 5 = 400 min - **Total B:** 2,000 + 400 = **2,400 min** **Machine C:** - M: 100 × 15 = 1,500 min - N: 80 × 5 = 400 min - **Total C:** 1,500 + 400 = **1,900 min** **Available time per week:** 2,400 min **Analysis:** - Machine A requires **2,700 min**, exceeding the available 2,400 min → **constraint** - Machine B is at capacity if producing max demand - Machine C has slack **Conclusion:** **Machine A** is the bottleneck (constraint). --- ### **b. Which product mix provides the highest gross profit?** **Step 1:** Calculate profit per unit (ignoring operating expenses): | Product | Selling Price | Raw Material Cost | Gross Profit per Unit | Profit per Min on A | |---|---|---|---|---| | M | $190 | $60 | $130 | $130 / 15 ≈ **$8.67** | | N | $200 | $40 + $55 = $95 | $105 | $105 / 15 ≈ **$7.00** | **Step 2:** To maximize profit within the constraint (Machine A time), produce as many units of the higher profit/min product (Product M): - Max demand for M: 100 units (requires 1,500 min on A) - Remaining A time: 2,400 - 1,500 = 900 min - Units of N that can be produced: 900 / 15 = 60 units **Product mix:** - **100 units of M** - **60 units of N** --- ### **c. What is the maximum weekly net profit?** **Step 1:** Calculate total revenue: - M: 100 × $190 = **$19,000** - N: 60 × $200 = **$12,000** - **Total revenue:** $19,000 + $12,000 = **$31,000** **Step 2:** Calculate raw material costs: - M: 100 × $60 = **$6,000** - N: 60 × ($40 + $55) = 60 × $95 = **$5,700** - **Total raw materials:** $6,000 + $5,700 = **$11,700** **Step 3:** Gross profit: - $31,000 - $11,700 = **$19,300** **Step 4:** Subtract operating expenses: - $19,300 - $12,000 = **$7,300** --- ### **Final Summary** | Question | Answer | |---|---| | **a. Which machine is the constraint?** | **Machine A** | | **b. Which product mix maximizes gross profit?** | **100 units of Product M and 60 units of Product N** | | **c. What is the maximum weekly net profit?** | **$7,300** | --- ### **Conclusion:** - The limiting resource is **Machine A**. - The optimal production plan is **100 units of M** and **60 units of N** per week. - The **maximum net profit** achievable with this plan is **$7,300**. If you'd like further clarification or additional analysis, please ask!