LOGO Production Expanded problem

Sample Solution

I'd be glad to help you with the LOGO Production Expanded problem and the Advertising Budget Optimization problem, but I cannot access attached files or confidential information. However, I can provide general guidance and explanations to address the issues:

Problem 1: LOGO Production Expanded with Fixed Costs

Modeling Approach:

1. Binary Variables: Introduce a binary variable X_ij for each product i and plant j. This variable takes a value of 1 if product i is produced at plant j, and 0 otherwise.

2. Fixed Cost Representation: Multiply the fixed cost ($100,000) by X_ij to represent the fixed cost incurred if product i is produced at plant j.

3. Objective Function: Modify the original objective function (minimizing production and inventory costs) to include the sum of fixed costs across all products and plants. This can be formulated as:

   Minimize: Production and Inventory Costs + Σ (Fixed Cost * X_ij) for all i and j
   

4. Constraint on Production Locations: Remove the original constraint that allows a product to be produced at only one plant.

Full Answer Section

Solving with Integer Programming:

• Use a spreadsheet solver with integer programming capabilities (e.g., Excel Solver with the GRG Nonlinear add-in) to find the optimal solution.
• Set the objective function cell as the target cell to minimize.
• Add constraints for production quantities, inventory levels, and the newly introduced binary variables.
• Set X_ij variables to binary (usually 0 or 1).

Problem 2: Advertising Budget Optimization with Sensitivity Analysis

Solving with Excel Solver:

1. Open the Advertising Budget Optimization spreadsheet.
2. Identify the cells containing:
   • Decision variables (e.g., amount spent on advertising for each product)
   • Objective function (e.g., total profit)
   • Constraints (e.g., advertising budget limitation)
3. Activate the Excel Solver add-in (usually under the "Data" tab).
4. Set the "Set Objective" to the cell containing the total profit formula (maximize or minimize).
5. Add constraints:
   • Select "Add" and define the advertising budget constraint by referencing the cells for individual product advertising and the total budget limit. Set the relationship to "<=" (less than or equal to).
   • Add any other constraints present in the problem.
6. Click "Solve." Solver will search for the optimal solution that maximizes profit within the budget constraint.

Sensitivity Analysis with Lagrangian Multiplier:

1. Lagrangian Multiplier: After solving with Solver, look for the "Sensitivity" report (may vary depending on the Solver version). This report will include a section on the "Constraint Cells." The value associated with the advertising budget constraint is the Lagrangian multiplier.
2. Interpretation: The Lagrangian multiplier represents the marginal change in the objective function (profit) for a unit increase in the corresponding constraint (advertising budget).
3. Increasing Budget by $10:
   • Increase the total advertising budget by $10 in the spreadsheet.
   • Re-run Solver. Note the change in optimal profit.
   • Compare this change to the Lagrangian multiplier's value. Ideally, the profit change should be close to the Lagrangian multiplier, indicating its validity in approximating the marginal effect.
4. Increasing Budget by $40,000:
   • Repeat the process with a $40,000 increase in the advertising budget.
   • Analyze the profit change and compare it to the Lagrangian multiplier's prediction. Again, a close match strengthens the concept of the Lagrangian multiplier.

Problem 3: Missing Information

I cannot provide specific guidance for Problem 3 as the attached picture is unavailable. If you can describe the picture or provide the problem statement, I might be able to offer assistance.

Remember that these are general guidelines. The specific implementation might vary depending on your spreadsheet software and the structure of your original problems.