Using Excel Solver for Optimization Problems

Excel Solver is a powerful add-in that can be used to solve a variety of optimization problems. It allows you to find the maximum or minimum value of a target cell (objective function) by changing the values of other cells (decision variables), subject to certain constraints. This guide will walk you through the process of using Excel Solver effectively.

1. Enabling the Solver Add-in

Before you can use Solver, you need to make sure it’s enabled in Excel. Here’s how:

1. Go to File > Options.
2. In the Excel Options window, click on Add-ins.
3. In the “Manage” dropdown at the bottom of the window, select Excel Add-ins and click Go…
4. In the Add-ins window, check the box next to Solver Add-in and click OK.

After enabling the add-in, you should find the Solver tool under the Data tab in the Excel ribbon, usually on the far right.

2. Setting Up Your Spreadsheet

The key to using Solver effectively is setting up your spreadsheet in a clear and organized manner. This typically involves the following elements:

• Decision Variables: These are the cells that Solver will change to find the optimal solution. They represent the variables you have control over in your problem. Designate specific cells for these variables and ideally give them descriptive names.
• Objective Function: This is the cell that contains the formula you want to maximize or minimize. It should be a function of the decision variables. This represents the target you want to optimize (e.g., profit, cost, output).
• Constraints: These are the limitations or restrictions on the values of the decision variables. Constraints can be inequalities (e.g., “production must be less than or equal to 100”) or equalities (e.g., “total expenses must equal budget”). They are also typically functions of the decision variables.

Example: Production Planning

Let’s say you want to determine the optimal production levels for two products (Product A and Product B) to maximize profit. You have constraints on resources (labor and materials).

Your spreadsheet might look like this:

3. Using the Solver Dialog Box

Once your spreadsheet is set up, you can use the Solver dialog box to define the optimization problem.

1. Go to the Data tab and click on Solver.
2. Set Objective: In the Solver Parameters dialog box, specify the cell containing your objective function (e.g., $B$8).
3. To: Select whether you want to Maximize, Minimize, or set the objective to a specific Value Of.
4. By Changing Variable Cells: Specify the cells containing your decision variables (e.g., $B$2:$B$3).
5. Subject to the Constraints: Click the Add button to add constraints. For each constraint:
   • Cell Reference: Select the cell containing the formula related to the constraint (e.g., $B$13 for labor hours used).
   • Operator: Choose the appropriate operator (<=, =, >=, int (integer), bin (binary), or dif (all different)).
   • Constraint: Enter the value or cell reference representing the constraint limit (e.g., $B$14 for available labor hours).
   • Click Add to add another constraint or OK to finish.
6. Select a Solving Method: From the “Select a Solving Method” dropdown, choose an appropriate method:
   • GRG Nonlinear: This is a good choice for most non-linear problems. It’s a gradient-based method that works well when the objective function and constraints are smooth and continuous.
   • Simplex LP: Use this for linear programming problems where both the objective function and constraints are linear. It’s generally faster and more reliable for linear problems.
   • Evolutionary: Use this for problems that are non-smooth, discontinuous, or highly complex. It’s a global optimization method that can find solutions even when other methods fail, but it can be slower.
7. Options: Click the Options button to adjust settings like the precision, convergence, and iteration limits. The default values are often sufficient, but you might need to adjust them for complex problems. Important options include:
   • Precision: Controls how accurately Solver tries to find the optimal solution. Lower values (e.g., 0.000001) mean higher precision but may take longer to solve.
   • Convergence: Defines how much the objective function can change before Solver stops.
   • Iterations: Sets a maximum number of iterations Solver will perform.
   • Assume Linear Model: Check this box if you know your problem is linear (for Simplex LP).
   • Assume Non-Negative: Check this box if your decision variables must be non-negative.
8. Click Solve.

4. Interpreting the Results

After Solver runs, it will display a Solver Results dialog box. This box provides information about whether Solver found a solution and what that solution is.

• Solver found a solution. All Constraints and Optimality conditions are satisfied: This indicates that Solver successfully found a solution that meets all the constraints.
• Solver could not find a feasible solution: This means that there is no solution that satisfies all the constraints. You might need to review your constraints and ensure they are not contradictory.
• Solver has converged to the current solution. All Constraints are satisfied: This suggests that Solver has found a local optimum. It’s possible that there’s a better solution elsewhere, especially for non-linear problems. Try different starting values for the decision variables.
• Solver found a solution, but some constraints may not be satisfied. This means that the solution is not perfectly optimal, and you should review the constraints and objective function.

The Solver Results dialog box also allows you to:

• Keep Solver Solution: This will update the values in your spreadsheet with the solution found by Solver.
• Restore Original Values: This will revert the values in your spreadsheet to what they were before Solver ran.
• Sensitivity Report, Limits Report, Answer Report: These reports provide valuable insights into the solution. The Sensitivity Report is particularly useful for understanding how changes in the constraints or objective function coefficients would affect the optimal solution.

5. Common Issues and Troubleshooting

• “Solver could not find a feasible solution”: This is a common issue. Double-check your constraints. Are they all necessary? Are they contradictory? Are you using the correct operators (<=, >=, =)? Make sure the units are consistent (e.g., don’t mix hours and minutes). Sometimes, a slight relaxation of a constraint can allow Solver to find a solution.
• “Solver has converged to the current solution”: This might indicate a local optimum. Try different starting values for the decision variables. For complex problems, the Evolutionary solving method may find a better global optimum.
• Slow Solving Time: Complex problems with many variables and constraints can take a long time to solve. Simplify the problem if possible, or adjust the Solver options (e.g., increase the iteration limit, adjust precision).
• Incorrect Results: Verify that your formulas are correct and that the objective function and constraints accurately represent the problem you are trying to solve. Double-check your decision variable cells are the ones you intend to change.

6. Beyond the Basics

Excel Solver is a versatile tool that can be applied to a wide range of problems, including:

• Linear Programming: Optimization problems with linear objective functions and linear constraints.
• Non-Linear Programming: Optimization problems with non-linear objective functions or constraints.
• Integer Programming: Optimization problems where the decision variables must be integers. This is achieved using the ‘int’ constraint type.
• Binary Programming: Optimization problems where the decision variables must be either 0 or 1. This is achieved using the ‘bin’ constraint type.
• Transportation Problems: Minimizing the cost of transporting goods from multiple sources to multiple destinations.
• Assignment Problems: Assigning tasks to individuals to maximize efficiency or minimize cost.
• Capital Budgeting: Selecting the best set of projects to invest in, given limited resources.

By understanding the fundamentals of Excel Solver and carefully setting up your spreadsheets, you can leverage its power to solve complex optimization problems and make better decisions.