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- Assignment Problem
Formulate the Model | Trial and Error | Solve the Model
Use the solver in Excel to find the assignment of persons to tasks that minimizes the total cost.
Formulate the Model
The model we are going to solve looks as follows in Excel.
1. To formulate this assignment problem , answer the following three questions.
a. What are the decisions to be made? For this problem, we need Excel to find out which person to assign to which task (Yes=1, No=0). For example, if we assign Person 1 to Task 1, cell C10 equals 1. If not, cell C10 equals 0.
b. What are the constraints on these decisions? Each person can only do one task (Supply=1). Each task only needs one person (Demand=1).
c. What is the overall measure of performance for these decisions? The overall measure of performance is the total cost of the assignment, so the objective is to minimize this quantity.
2. To make the model easier to understand, create the following named ranges .
3. Insert the following functions.
Explanation: The SUM functions calculate the number of tasks assigned to a person and the number of persons assigned to a task. Total Cost equals the sumproduct of Cost and Assignment.
Trial and Error
With this formulation, it becomes easy to analyze any trial solution.
For example, if we assign Person 1 to Task 1, Person 2 to task 2 and Person 3 to Task 3, Tasks Assigned equals Supply and Persons Assigned equals Demand. This solution has a total cost of 147.
It is not necessary to use trial and error. We shall describe next how the Excel Solver can be used to quickly find the optimal solution.
Solve the Model
To find the optimal solution, execute the following steps.
1. On the Data tab, in the Analyze group, click Solver.
Note: can't find the Solver button? Click here to load the Solver add-in .
Enter the solver parameters (read on). The result should be consistent with the picture below.
You have the choice of typing the range names or clicking on the cells in the spreadsheet.
2. Enter TotalCost for the Objective.
3. Click Min.
4. Enter Assignment for the Changing Variable Cells.
5. Click Add to enter the following constraint.
Note: binary variables are either 0 or 1.
6. Click Add to enter the following constraint.
7. Click Add to enter the following constraint.
8. Check 'Make Unconstrained Variables Non-Negative' and select 'Simplex LP'.
9. Finally, click Solve.
The optimal solution:
Conclusion: it is optimal to assign Person 1 to task 2, Person 2 to Task 3 and Person 3 to Task 1. This solution gives the minimum cost of 129. All constraints are satisfied.
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Balanced Assignment Problem Using Excel Solver
In this post, you’ll learn about Assignment problem and know how to solve a assignment problem using Excel Solver Add-In .
Assignment problem
An Assignment problem is a type of linear programming problem, which is about assigning the correct number of resources to various destinations or problems. This gives you the optimal solution which is either maximizing the profit or minimizing the cost of production .
How to Solve Balanced Assignment Problem Using Excel Solver Add-In?
Let’s look at the range of values in the screenshot attached below. We need to minimize the cost of production in this problem.
The Assigned values are the sum of the respective Customer value row, similarly the sum of Task columns for the Assigned. So use the =SUM() formula.
For finding the Optimal solution use the function =SUMPRODUCT() .
- After you’re done entering the values, go to the Data tab and under Analyze group select the Solver tool .
- Set the objective in the correct cell. Select Min in the To field.
- Changing variables are the tasks to the respective customer whose values aren’t determined still.
- The Constraints are Assigned = Supply , Assigned = Demand, and the range of the changing variables are binary .
- Finally select the solving method as Simplex LP and click on Solve .
Click on OK to get the solution.
A optimal solution is obtained.
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Solving an Assignment Problem
This section presents an example that shows how to solve an assignment problem using both the MIP solver and the CP-SAT solver.
In the example there are five workers (numbered 0-4) and four tasks (numbered 0-3). Note that there is one more worker than in the example in the Overview .
The costs of assigning workers to tasks are shown in the following table.
The problem is to assign each worker to at most one task, with no two workers performing the same task, while minimizing the total cost. Since there are more workers than tasks, one worker will not be assigned a task.
MIP solution
The following sections describe how to solve the problem using the MPSolver wrapper .
Import the libraries
The following code imports the required libraries.
Create the data
The following code creates the data for the problem.
The costs array corresponds to the table of costs for assigning workers to tasks, shown above.
Declare the MIP solver
The following code declares the MIP solver.
Create the variables
The following code creates binary integer variables for the problem.
Create the constraints
Create the objective function.
The following code creates the objective function for the problem.
The value of the objective function is the total cost over all variables that are assigned the value 1 by the solver.
Invoke the solver
The following code invokes the solver.
Print the solution
The following code prints the solution to the problem.
Here is the output of the program.
Complete programs
Here are the complete programs for the MIP solution.
CP SAT solution
The following sections describe how to solve the problem using the CP-SAT solver.
Declare the model
The following code declares the CP-SAT model.
The following code sets up the data for the problem.
The following code creates the constraints for the problem.
Here are the complete programs for the CP-SAT solution.
Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 License , and code samples are licensed under the Apache 2.0 License . For details, see the Google Developers Site Policies . Java is a registered trademark of Oracle and/or its affiliates.
Last updated 2023-01-02 UTC.
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Transportation and Assignment Problems
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