Let's break down the problem step by step: ### 1. **Define Decision Variables** Let: - \( x_i \) = number of units of product \( i \) produced (\( i = 1, 2, 3 \)) - \( y_i \) = 1 if any quantity of product \( i \) is produced, 0 otherwise (\( i = 1, 2, 3 \)) ### 2. **Labor Hour Requirements** | Dept. | Product 1 | Product 2 | Product 3 | Available Hours | |-------|-----------|-----------|-----------|----------------| | A | 1.50 | 3.00 | 2.00 | 450 | | B | 2.00 | 1.00 | 2.50 | 350 | | C | 0.25 | 0.25 | 0.50 | 50 | ### 3. **Profit Contributions** - Product 1: $25/unit - Product 2: $28/unit - Product 3: $30/unit ### 4. **Setup Costs** - Product 1: $370 (only if any produced, i.e., if \( y_1 = 1 \)) - Product 2: $210 (if \( y_2 = 1 \)) - Product 3: $310 (if \( y_3 = 1 \)) ### 5. **Maximum Realized Total Setup Cost** \[ 370y_1 + 210y_2 + 310y_3 \leq 185 \] ### 6. **Constraints** #### a. **Labor Hours** \[ \begin{align*} 1.5x_1 + 3x_2 + 2x_3 &\leq 450 \quad \text{(Department A)} \\ 2x_1 + x_2 + 2.5x_3 &\leq 350 \quad \text{(Department B)} \\ 0.25x_1 + 0.25x_2 + 0.5x_3 &\leq 50 \quad \text{(Department C)} \\ \end{align*} \] #### b. **Linking Constraints (Setup Costs)** \[ x_i \leq M_i y_i \quad \text{for } i = 1,2,3 \] Where \( M_1, M_2, M_3 \) are large enough upper bounds for each product. #### c. **Product 1 Maximum Production if Setup Paid** If setup is paid for product 1 (\( y_1 = 1 \)), then: \[ x_1 \leq 185 \] #### d. **Setup Cost Constraint** \[ 370y_1 + 210y_2 + 310y_3 \leq 185 \] ### 7. **Objective Function** \[ \text{Maximize: } Z = 25x_1 + 28x_2 + 30x_3 - 370y_1 - 210y_2 - 310y_3 \] --- ## **Formulation** \[ \begin{align*} \text{Maximize:} \quad & 25x_1 + 28x_2 + 30x_3 - 370y_1 - 210y_2 - 310y_3 \\ \text{Subject to:} \\ & 1.5x_1 + 3x_2 + 2x_3 \leq 450 \\ & 2x_1 + x_2 + 2.5x_3 \leq 350 \\ & 0.25x_1 + 0.25x_2 + 0.5x_3 \leq 50 \\ & x_1 \leq 185y_1 \\ & x_2 \leq M_2 y_2 \\ & x_3 \leq M_3 y_3 \\ & 370y_1 + 210y_2 + 310y_3 \leq 185 \\ & x_1, x_2, x_3 \geq 0 \\ & y_1, y_2, y_3 \in \{0, 1\} \end{align*} \] --- ## **Step-by-Step Solution** ### Step 1: **Setup Cost Constraint Analysis** Since setup costs must be ≤ 185, test which products can be selected: - Product 1: 370 (too high alone) - Product 2: 210 (too high alone) - Product 3: 310 (too high alone) - Any combination of products: All combinations exceed 185. ### **Conclusion** **No product can be produced** because the setup cost for any single product (or combination) **exceeds the allowed maximum setup cost of 185**. --- ## **Final Answer** \[ \boxed{ \begin{align*} x_1 &= 0 \\ x_2 &= 0 \\ x_3 &= 0 \\ y_1 &= 0 \\ y_2 &= 0 \\ y_3 &= 0 \\ \text{Maximum Profit} &= 0 \end{align*} } \] **Explanation:** No product can be produced because the setup cost constraint is too restrictive. The minimum setup cost for any product is 210, which is above the maximum allowed of 185. Thus, the optimal solution is to produce nothing, yielding zero profit.

Let's analyze the problem carefully and go through each step in detail, including all calculations to arrive at the final solution. --- ## **Problem Recap** - **Decision variables:** - \( x_i \): units of product \( i \) produced (\( i=1,2,3 \)) - \( y_i \): binary, 1 if product \( i \) is produced, 0 if not - **Profit per unit:** - Product 1: \$25 - Product 2: \$28 - Product 3: \$30 - **Setup costs:** - Product 1: \$370 - Product 2: \$210 - Product 3: \$310 - **Available labor hours:** | Department | Product 1 | Product 2 | Product 3 | Hours available | |--------------|------------|------------|------------|----------------| | A | 1.50 | 3.00 | 2.00 | 450 | | B | 2.00 | 1.00 | 2.50 | 350 | | C | 0.25 | 0.25 | 0.50 | 50 | - **Total maximum allowable setup cost sum:** \$185 --- ## **Step 1: Write the constraints** ### Labor constraints: \[ \begin{cases} 1.5x_1 + 3x_2 + 2x_3 \leq 450 \quad \text{(Dept A)} \\ 2x_1 + x_2 + 2.5x_3 \leq 350 \quad \text{(Dept B)} \\ 0.25x_1 + 0.25x_2 + 0.5x_3 \leq 50 \quad \text{(Dept C)} \\ \end{cases} \] ### Setup cost constraint: \[ 370 y_1 + 210 y_2 + 310 y_3 \leq 185 \] ### Linking constraints: \[ x_i \leq M_i y_i, \quad i=1,2,3 \] where \( M_i \) are large enough upper bounds for production. --- ## **Step 2: Analyze the setup cost constraint** Since each product's setup cost exceeds \$185 individually: - \( 370 y_1 \leq 185 \Rightarrow y_1=0 \) (since 370 > 185) - \( 210 y_2 \leq 185 \Rightarrow y_2=0 \) - \( 310 y_3 \leq 185 \Rightarrow y_3=0 \) Thus, **none** of the products can be produced because even setting up one product exceeds the maximum total setup cost allowed. --- ## **Step 3: Final check** - Producing any product will cost at least \$210 (product 2), \$310 (product 3), or \$370 (product 1), all exceeding the setup cost limit of \$185. - Therefore, the only feasible solution is **not producing any products**. --- ## **Step 4: Compute the profit** - Since no products are produced: \[ x_1 = x_2 = x_3 = 0 \] - Total profit: \[ Z = 25 \times 0 + 28 \times 0 + 30 \times 0 = 0 \] --- ## **Final Answer** \[ \boxed{ \begin{aligned} & \textbf{Produce nothing} \\ & x_1 = 0, \quad x_2=0, \quad x_3=0 \\ & \text{Total profit} = \boxed{0} \end{aligned} } \] --- ## **Summary** - The setup costs are too high relative to the maximum allowed setup budget (\$185). - **No products can be produced**. - The optimal profit is **zero**. --- **Note:** If the maximum setup cost constraint were higher, or if the problem allowed for different constraints, the solution approach would involve solving the LP with feasible \( y_i \) and \( x_i \). But under current constraints, producing nothing is optimal.