Let's solve the problem step by step: ### 1. **Define Decision Variables** Let: - \( x_1 \) = Number of Product A produced (0–40 units, cost $10/unit) - \( x_2 \) = Number of Product A produced (41–100 units, cost $9/unit) - \( x_3 \) = Number of Product A produced (101–150 units, cost $7/unit) - \( y_1 \) = Number of Product B produced (0–50 units, cost $6/unit) - \( y_2 \) = Number of Product B produced (51–100 units, cost $5/unit) - \( z_1 \) = Number of Product C produced (0–100 units, cost $5/unit) - \( z_2 \) = Number of Product C produced (over 100 units, cost $4/unit) So, - Total Product A produced = \( x_1 + x_2 + x_3 \) - Total Product B produced = \( y_1 + y_2 \) - Total Product C produced = \( z_1 + z_2 \) --- ### 2. **Objective Function (Maximize Profit)** **Profit = Sales Revenue - Production Cost** - Selling price: A = $12, B = $9, C = $7 So, \[ \text{Revenue} = 12(x_1 + x_2 + x_3) + 9(y_1 + y_2) + 7(z_1 + z_2) \] \[ \text{Cost} = 10x_1 + 9x_2 + 7x_3 + 6y_1 + 5y_2 + 5z_1 + 4z_2 \] Therefore, \[ \text{Maximize } Z = [12(x_1 + x_2 + x_3) + 9(y_1 + y_2) + 7(z_1 + z_2)] - [10x_1 + 9x_2 + 7x_3 + 6y_1 + 5y_2 + 5z_1 + 4z_2] \] --- ### 3. **Constraints** **a) Production Limits (from cost table):** \[ 0 \leq x_1 \leq 40 \] \[ 0 \leq x_2 \leq 60 \quad (since 41-100 means up to 60 units) \] \[ 0 \leq x_3 \leq 50 \quad (since 101-150 means up to 50 units) \] \[ 0 \leq y_1 \leq 50 \] \[ 0 \leq y_2 \leq 50 \quad (since 51-100 means up to 50 units) \] \[ 0 \leq z_1 \leq 100 \] \[ z_2 \geq 0 \] --- **b) Resource Constraints** | Resource | Product A | Product B | Product C | Available | |-----------------|--------------------|------------------|---------------|-------------| | Engineering | 1 per unit | 2 per unit | 1 per unit | 100 hours | | Direct Labor | 10 per unit | 4 per unit | 5 per unit | 700 hours | | Material | 3 per unit | 2 per unit | 1 per unit | 400 lbs | So, for all units: \[ (x_1 + x_2 + x_3) + 2(y_1 + y_2) + (z_1 + z_2) \leq 100 \] \[ 10(x_1 + x_2 + x_3) + 4(y_1 + y_2) + 5(z_1 + z_2) \leq 700 \] \[ 3(x_1 + x_2 + x_3) + 2(y_1 + y_2) + (z_1 + z_2) \leq 400 \] --- ### 4. **Non-negativity Constraints** \[ x_1, x_2, x_3, y_1, y_2, z_1, z_2 \geq 0 \] --- ### 5. **Excel Formulation Steps** **Set up cells for:** - \( x_1, x_2, x_3, y_1, y_2, z_1, z_2 \) (your decision variables) - Compute total production for each product - Compute total resource usage for each resource - Compute total revenue and total cost - Compute profit (objective function) **Use Solver:** - Set objective to maximize profit - By changing \( x_1, x_2, x_3, y_1, y_2, z_1, z_2 \) - Subject to the constraints listed above --- ### **Summary** **Decision Variables:** \( x_1, x_2, x_3, y_1, y_2, z_1, z_2 \) **Objective Function:** \[ \text{Maximize } Z = [12(x_1 + x_2 + x_3) + 9(y_1 + y_2) + 7(z_1 + z_2)] - [10x_1 + 9x_2 + 7x_3 + 6y_1 + 5y_2 + 5z_1 + 4z_2] \] **Constraints:** \[ \begin{align*} x_1 + x_2 + x_3 + 2(y_1 + y_2) + z_1 + z_2 &\leq 100 \\ 10(x_1 + x_2 + x_3) + 4(y_1 + y_2) + 5(z_1 + z_2) &\leq 700\\ 3(x_1 + x_2 + x_3) + 2(y_1 + y_2) + z_1 + z_2 &\leq 400\\ 0 \leq x_1 \leq 40, \quad 0 \leq x_2 \leq 60, \quad 0 \leq x_3 \leq 50 \\ 0 \leq y_1 \leq 50, \quad 0 \leq y_2 \leq 50 \\ 0 \leq z_1 \leq 100, \quad z_2 \geq 0 \\ x_1, x_2, x_3, y_1, y_2, z_1, z_2 \geq 0 \\ \end{align*} \] --- **To solve in Excel:** 1. Enter the variables in cells. 2. Compute all components (resources, total units, cost, revenue, profit). 3. Use Solver to maximize profit subject to the constraints. --- **If you need the actual Excel layout or optimal solution, let me know!**

Certainly! Let's walk through the detailed step-by-step process to formulate and solve the problem, including all calculations, assumptions, and how the given data is used. --- ## **Step 1: Understanding the Data and Requirements** ### **Products:** - A, B, C - Each has a piecewise production cost depending on the units produced - Selling prices are fixed: - A = $12/unit - B = $9/unit - C = $7/unit ### **Available Resources:** - Engineering hours: 100 hours - Direct labor hours: 700 hours - Material: 400 lbs ### **Per-unit resource consumption:** | Resource | Product A | Product B | Product C | |------------|--------------|--------------|--------------| | Engineering hours | 1 hr | 2 hrs | 1 hr | | Direct labor hours | 10 hrs | 4 hrs | 5 hrs | | Material | 3 lbs | 2 lbs | 1 lb | --- ## **Step 2: Define Decision Variables** Because of the piecewise cost structure, we split production quantities into segments for each product. ### **Product A:** - \( x_1 \): units produced in 0–40 units (cost $10/unit) - \( x_2 \): units produced in 41–100 units (cost $9/unit) - \( x_3 \): units produced in 101–150 units (cost $7/unit) Total units of A: \( X_A = x_1 + x_2 + x_3 \) ### **Product B:** - \( y_1 \): units in 0–50 (cost $6/unit) - \( y_2 \): units in 51–100 (cost $5/unit) Total units of B: \( X_B = y_1 + y_2 \) ### **Product C:** - \( z_1 \): units in 0–100 (cost $5/unit) - \( z_2 \): units over 100 (cost $4/unit) Total units of C: \( X_C = z_1 + z_2 \) --- ## **Step 3: Write the Objective Function** ### **Sales Revenue:** \[ \text{Revenue} = 12X_A + 9X_B + 7X_C \] ### **Production Cost:** - For A: \[ \text{Cost}_A = 10x_1 + 9x_2 + 7x_3 \] - For B: \[ \text{Cost}_B = 6y_1 + 5y_2 \] - For C: \[ \text{Cost}_C = 5z_1 + 4z_2 \] ### **Profit:** \[ \boxed{ \text{Maximize } Z = (12X_A + 9X_B + 7X_C) - (10x_1 + 9x_2 + 7x_3 + 6y_1 + 5y_2 + 5z_1 + 4z_2) } \] --- ## **Step 4: Write Constraints** ### **a) Production segment constraints:** - \( x_1 \geq 0, x_2 \geq 0, x_3 \geq 0 \) - \( y_1 \geq 0, y_2 \geq 0 \) - \( z_1 \geq 0, z_2 \geq 0 \) ### **b) Production capacity limits:** From the piecewise costs: | Product | Segment 1 (units) | Segment 2 (units) | Segment 3 (units) | |---------|-------------------|-------------------|-------------------| | A | \( 0 \leq x_1 \leq 40 \) | \( 0 \leq x_2 \leq 60 \) | \( 0 \leq x_3 \leq 50 \) | | B | \( 0 \leq y_1 \leq 50 \) | \( 0 \leq y_2 \leq 50 \) | N/A | | C | \( 0 \leq z_1 \leq 100 \) | \( z_2 \geq 0 \) | N/A | --- ### **c) Resource constraints:** Using per-unit resource consumption: - **Engineering hours:** \[ 1 \times x_1 + 1 \times x_2 + 1 \times x_3 + 2 \times y_1 + 2 \times y_2 + 1 \times z_1 + 1 \times z_2 \leq 100 \] - **Direct labor hours:** \[ 10x_1 + 9x_2 + 7x_3 + 4y_1 + 5y_2 + 5z_1 + 4z_2 \leq 700 \] - **Material:** \[ 3x_1 + 2x_2 + 1x_3 + 2y_1 + 2y_2 + 1z_1 + 1z_2 \leq 400 \] --- ## **Step 5: Summarize the complete LP model** ### **Decision variables:** \[ x_1, x_2, x_3, y_1, y_2, z_1, z_2 \geq 0 \] ### **Objective:** \[ \boxed{ \text{Maximize } Z = (12(x_1 + x_2 + x_3) + 9(y_1 + y_2) + 7(z_1 + z_2)) - (10x_1 + 9x_2 + 7x_3 + 6y_1 + 5y_2 + 5z_1 + 4z_2) } \] which simplifies to: \[ Z = (12x_1 + 12x_2 + 12x_3 + 9y_1 + 9y_2 + 7z_1 + 7z_2) - (10x_1 + 9x_2 + 7x_3 + 6y_1 + 5y_2 + 5z_1 + 4z_2) \] \[ Z = (12x_1 - 10x_1) + (12x_2 - 9x_2) + (12x_3 - 7x_3) + (9y_1 - 6y_1) + (9y_2 - 5y_2) + (7z_1 - 5z_1) + (7z_2 - 4z_2) \] \[ Z = 2x_1 + 3x_2 + 5x_3 + 3y_1 + 4y_2 + 2z_1 + 3z_2 \] --- ## **Step 6: Final LP in mathematical form** \[ \begin{cases} \text{Maximize } Z = 2x_1 + 3x_2 + 5x_3 + 3y_1 + 4y_2 + 2z_1 + 3z_2 \\ \text{subject to} \\ x_1 \leq 40 \\ x_2 \leq 60 \\ x_3 \leq 50 \\ y_1 \leq 50 \\ y_2 \leq 50 \\ z_1 \leq 100 \\ x_1, x_2, x_3, y_1, y_2, z_1, z_2 \geq 0 \\ \text{Resource constraints as above} \end{cases} \] --- ## **Step 7: Solving the LP** ### **Using Excel or Solver:** - Input decision variables - Set the objective function (profit) - Add the constraints - Run Solver to find the optimal production quantities. --- ## **Summary:** - **The key step** was to split production into segments because costs depend on production levels. - **The profit** is maximized by choosing the optimal number of units in each segment, respecting resource constraints and segment capacities. - **The final step** involves solving the LP with software tools like Excel Solver. --- ## **Note:** Because the question asks for detailed calculations, **here's an example of how to compute resource consumption and profit for a hypothetical production plan:** **Suppose:** \[ x_1=30,\quad x_2=20,\quad x_3=10,\quad y_1=40,\quad y_2=10,\quad z_1=80,\quad z_2=20 \] **Calculate total units:** \[ X_A = 30 + 20 + 10 = 60 \] \[ X_B = 40 + 10 = 50 \] \[ X_C = 80 + 20 = 100 \] **Calculate resource usage:** - **Engineering hours:** \[ (1 \times 30) + (1 \times 20) + (1 \times 10) + (2 \times 40) + (2 \times 10) + (1 \times 80) + (1 \times 20) = 30 + 20 + 10 + 80 + 20 + 80 + 20 = 260 \text{ hours} \] - **Direct labor hours:** \[ 10 \times 30 + 9 \times 20 + 7 \times 10 + 4 \times 40 + 5 \times 10 + 5 \times 80 + 4 \times 20 = 300 + 180 + 70 + 160 + 50 + 400 + 80 = 1,290 \text{ hours} \] - **Material:** \[ 3 \times 30 + 2 \times 20 + 1 \times 10 + 2 \times 40 + 2 \times 10 + 1 \times 80 + 1 \times 20 = 90 + 40 + 10 + 80 + 20 + 80 + 20 = 340 \text{ lbs} \] **Check resource limits:** - Engineering: 260 hours ≤ 100 hours → **Not feasible**, so this plan is invalid. **Calculate profit:** \[ \text{Revenue} = 12 \times 60 + 9 \times 50 + 7 \times 100 = 720 + 450 + 700 = 1,870 \] \[ \text{Cost} = (10 \times 30 + 9 \times 20 + 7 \times 10) + (6 \times 40 + 5 \times 10) + (5 \times 80 + 4 \times 20) = (300 + 180 + 70) + (240 + 50) + (400 + 80) = 550 + 290 + 480 = 1,320 \] \[ \text{Profit} = 1,870 - 1,320 = 550 \] --- ## **Final Notes:** - Actual optimal solution requires solving the LP with constraints. The example above shows the process. - Use Excel Solver or similar for the exact optimal production plan. --- **Let me know if you'd like me to help set up the Excel sheet or interpret Solver results!**