Let's break down the problem step by step, formulate the linear programming model, and provide explanations and a conclusion. --- ## **Step 1: Define the Decision Variables** As given: - \( X_1 \): Number of city Democrats interviewed - \( X_2 \): Number of city Republicans interviewed - \( X_3 \): Number of city Independents interviewed - \( X_4 \): Number of suburban Democrats interviewed - \( X_5 \): Number of suburban Republicans interviewed - \( X_6 \): Number of suburban Independents interviewed --- ## **Step 2: Formulate the Objective Function** The goal is to **minimize the total cost** of interviews. **Costs per interview:** - City: D = $18, R = $16, I = $22 - Suburbs: D = $21, R = $20, I = $25 So, the objective function to minimize (Z): \[ Z = 18X_1 + 16X_2 + 22X_3 + 21X_4 + 20X_5 + 25X_6 \] --- ## **Step 3: Identify the Constraints** The problem mentions constraints but the details are missing in the image. Normally, constraints might include: - Minimum or maximum interviews per group - Total voters to be interviewed - Proportional representation Since these are not detailed in the image, let's focus on the provided questions, which reference the results from Excel Solver. --- ## **Step 4: Analyze the Questions** ### **Question 12: Range for \( X_1 \)** - At the optimal solution, what is the value (range) for \( X_1 \) (City Democrats interviewed)? ### **Question 13: Range for Objective Function \( Z \)** - What is the value (range) for the total cost \( Z \)? --- ## **Step 5: Answering the Questions** The answers to 12 and 13 depend on running the actual optimization in Excel Solver, which isn't possible in this text format. However, here's the process you would follow: ### **How to Solve with Excel Solver:** 1. **Enter Decision Variables:** Input \( X_1 \) through \( X_6 \) as changing cells. 2. **Objective Function:** Define cell for \( Z \) using the cost formula. 3. **Constraints:** Add any constraints as per the contract requirements. 4. **Solver Settings:** Set Solver to minimize \( Z \) by changing \( X_1 \)–\( X_6 \), subject to constraints. 5. **Solve:** Run Solver to get optimal values. ### **Based on the Typical Output:** Without explicit constraints, let's guess likely answers based on the cost structure (city interviews are generally cheaper than suburbs, especially for Democrats and Republicans): - **City Democrats (\( X_1 \))**: Since city Democrats are relatively cheap, the model will likely maximize city Democrats within the allowed constraints. - **Objective Function (\( Z \))**: The minimum possible cost is likely more than $10,000 but probably less than $20,000 for a reasonable number of interviews. #### **Therefore, likely answers:** **Question 12:** The correct answer is probably **d. 400 < X₁ ≤ 500** or **c. 500 < X₁ ≤ 600** (if constraints allow a large number). **Question 13:** The correct answer is probably **e. $15,000 < Z ≤ $20,000** or **d. $10,000 < Z ≤ $15,000** (for typical marketing surveys). --- ## **Step 6: Conclusion** **Conclusion:** To find the exact values, you'd need to input all constraints and costs into Excel Solver and solve. The process involves formulating the objective function to minimize cost, defining the variables for each voter type/geography, and applying any contractual constraints. The solution will give you both the optimal number of each voter type to interview and the minimum total cost. **In summary:** - **Formulate the LP model** - **Input into Excel Solver** - **Find values for \( X_1 \) and \( Z \) from the Solver output** - **Choose the answer that matches Solver's output** If you provide the specific constraints, I can help formulate the full LP model and walk through a sample solution!