Let's break down the problem step by step: --- ### 1. **Data Extraction** #### **Product Details** | Product | Selling Price | Max Demand (units/week) | |----------|--------------|-------------------------| | M | $190 | 100 | | N | $200 | 80 | #### **Raw Material Costs** - Product M: RM-1 ($60/unit), RM-2 ($40/unit) - Product N: RM-2 ($40/unit), RM-3 ($55/unit) #### **Machine Times Per Unit** | Machine | Product M (min/unit) | Product N (min/unit) | |-----------|---------------------|----------------------| | **A** | 15 | 15 | | **B** | 20 | 5 | | **C** | 0 | 5 | #### **Capacity** - Each machine: 2,400 min/week #### **Operating Expenses:** $12,000/week (exclude raw materials) --- ## **a. Which machine is the constraint?** Calculate maximum units each machine can handle for each product: ### **For 100 M and 80 N units:** #### **Machine A** - M: 100 × 15 = 1,500 min - N: 80 × 15 = 1,200 min - **Total:** 2,700 min (**Exceeds 2,400 min**) #### **Machine B** - M: 100 × 20 = 2,000 min - N: 80 × 5 = 400 min - **Total:** 2,400 min (**Just at limit**) #### **Machine C** - M: 100 × 0 = 0 min - N: 80 × 5 = 400 min - **Total:** 400 min **Machine A requires 2,700 min for full demand, but only 2,400 min are available. This is the first machine to hit its limit (the constraint).** ### **Answer a:** **Constraint:** **Machine A** --- ## **b. Which product mix provides the highest gross profit?** ### **Step 1: Calculate Gross Profit per Unit** #### **Product M** - Selling price: $190 - Raw materials: $60 (RM-1) + $40 (RM-2) = $100 - **Gross profit/unit:** $190 - $100 = **$90** #### **Product N** - Selling price: $200 - Raw materials: $40 (RM-2) + $55 (RM-3) = $95 - **Gross profit/unit:** $200 - $95 = **$105** ### **Step 2: Maximize Gross Profit Within Constraints** Let - \( x \) = units of M per week - \( y \) = units of N per week #### **Constraints:** - \( x \leq 100 \) - \( y \leq 80 \) - **Machine A:** \( 15x + 15y \leq 2400 \) #### **Objective:** Maximize \( 90x + 105y \) #### **Solve the constraints:** From machine A: \( 15x + 15y \leq 2400 \implies x + y \leq 160 \) But also, \( x \leq 100 \), \( y \leq 80 \) So, possible combinations: - If we do all M: \( x = 100, y = 0 \) (Total = 100) - If all N: \( y = 80, x = 0 \) (Total = 80) - If mix: \( x + y = 160 \), subject to \( x \leq 100, y \leq 80 \) So, try \( x = 100 \Rightarrow y = 60 \) (since 100 + 60 = 160, and y ≤ 80) Try \( y = 80 \Rightarrow x = 80 \) (since 80 + 80 = 160, and x ≤ 100) Calculate gross profit for both: - \( x = 100, y = 60 \): \( 90 \times 100 + 105 \times 60 = 9000 + 6300 = 15,300 \) - \( x = 80, y = 80 \): \( 90 \times 80 + 105 \times 80 = 7,200 + 8,400 = 15,600 \) **So, the highest gross profit is with x = 80, y = 80.** ### **Answer b:** **Product mix:** - **Product M units:** 80 - **Product N units:** 80 --- ## **c. What is the maximum weekly net profit?** - From above: 80 units M, 80 units N ### **Total Revenue:** - \( 80 \times 190 + 80 \times 200 = 15,200 + 16,000 = \$31,200 \) ### **Raw Material Cost:** - M: \( 80 \times (60 + 40) = 80 \times 100 = \$8,000 \) - N: \( 80 \times (40 + 55) = 80 \times 95 = \$7,600 \) - **Total Raw Material Cost:** \$8,000 + \$7,600 = \$15,600 ### **Operating Expense:** \$12,000 ### **Net Profit:** \[ \text{Net Profit} = \text{Total Revenue} - \text{Raw Materials} - \text{Operating Expense} \] \[ = \$31,200 - \$15,600 - \$12,000 = \boxed{\$3,600} \] --- ## **Summary Table** | Part | Answer | |---|---| | a | **Machine A** is the constraint | | b | **Product mix:** 80 units M, 80 units N | | c | **Maximum weekly net profit:** **\$3,600** | --- ### **Final Answers** **a.** Machine A **b.** 80 units of Product M, 80 units of Product N **c.** Maximum weekly net profit: **\$3,600**

a. The constraint in the plant is Machine A, as it reaches its capacity limit first when producing the demand. b. The product mix that yields the highest gross profit is producing 80 units each of Products M and N. c. The maximum weekly net profit achievable with this product mix is \$3,600 after accounting for raw materials and operating expenses.