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Hungarian Maximum Matching Algorithm

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The Hungarian matching algorithm , also called the Kuhn-Munkres algorithm, is a \(O\big(|V|^3\big)\) algorithm that can be used to find maximum-weight matchings in bipartite graphs , which is sometimes called the assignment problem . A bipartite graph can easily be represented by an adjacency matrix , where the weights of edges are the entries. Thinking about the graph in terms of an adjacency matrix is useful for the Hungarian algorithm.

A matching corresponds to a choice of 1s in the adjacency matrix, with at most one 1 in each row and in each column.

The Hungarian algorithm solves the following problem:

In a complete bipartite graph \(G\), find the maximum-weight matching. (Recall that a maximum-weight matching is also a perfect matching.)

This can also be adapted to find the minimum-weight matching.

Say you are having a party and you want a musician to perform, a chef to prepare food, and a cleaning service to help clean up after the party. There are three companies that provide each of these three services, but one company can only provide one service at a time (i.e. Company B cannot provide both the cleaners and the chef). You are deciding which company you should purchase each service from in order to minimize the cost of the party. You realize that is an example of the assignment problem, and set out to make a graph out of the following information: \(\quad\) Company\(\quad\) \(\quad\) Cost for Musician\(\quad\) \(\quad\) Cost for Chef\(\quad\) \(\quad\) Cost for Cleaners\(\quad\) \(\quad\) Company A\(\quad\) \(\quad\) $108\(\quad\) \(\quad\) $125\(\quad\) \(\quad\) $150\(\quad\) \(\quad\) Company B\(\quad\) \(\quad\) $150\(\quad\) \(\quad\) $135\(\quad\) \(\quad\) $175\(\quad\) \(\quad\) Company C\(\quad\) \(\quad\) $122\(\quad\) \(\quad\) $148\(\quad\) \(\quad\) $250\(\quad\) Can you model this table as a graph? What are the nodes? What are the edges? Show Answer The nodes are the companies and the services. The edges are weighted by the price.

What are some ways to solve the problem above? Since the table above can be thought of as a \(3 \times 3\) matrix, one could certainly solve this problem using brute force, checking every combination and seeing what yields the lowest price. However, there are \(n!\) combinations to check, and for large \(n\), this method becomes very inefficient very quickly.

The Hungarian Algorithm Using an Adjacency Matrix

The hungarian algorithm using a graph.

With the cost matrix from the example above in mind, the Hungarian algorithm operates on this key idea: if a number is added to or subtracted from all of the entries of any one row or column of a cost matrix, then an optimal assignment for the resulting cost matrix is also an optimal assignment for the original cost matrix.

The Hungarian Method [1] Subtract the smallest entry in each row from all the other entries in the row. This will make the smallest entry in the row now equal to 0. Subtract the smallest entry in each column from all the other entries in the column. This will make the smallest entry in the column now equal to 0. Draw lines through the row and columns that have the 0 entries such that the fewest lines possible are drawn. If there are \(n\) lines drawn, an optimal assignment of zeros is possible and the algorithm is finished. If the number of lines is less than \(n\), then the optimal number of zeroes is not yet reached. Go to the next step. Find the smallest entry not covered by any line. Subtract this entry from each row that isn’t crossed out, and then add it to each column that is crossed out. Then, go back to Step 3.

Solve for the optimal solution for the example in the introduction using the Hungarian algorithm described above. Here is the initial adjacency matrix: Subtract the smallest value in each row from the other values in the row: Now, subtract the smallest value in each column from all other values in the column: Draw lines through the row and columns that have the 0 entries such that the fewest possible lines are drawn: There are 2 lines drawn, and 2 is less than 3, so there is not yet the optimal number of zeroes. Find the smallest entry not covered by any line. Subtract this entry from each row that isn’t crossed out, and then add it to each column that is crossed out. Then, go back to Step 3. 2 is the smallest entry. First, subtract from the uncovered rows: Now add to the covered columns: Now go back to step 3, drawing lines through the rows and columns that have 0 entries: There are 3 lines (which is \(n\)), so we are done. The assignment will be where the 0's are in the matrix such that only one 0 per row and column is part of the assignment. Replace the original values: The Hungarian algorithm tells us that it is cheapest to go with the musician from company C, the chef from company B, and the cleaners from company A. We can verify this by brute force. 108 + 135 + 250 = 493 108 + 148 + 175 = 431 150 + 125 + 250 = 525 150 + 148 + 150 = 448 122 + 125 + 175 = 422 122 + 135 + 150 = 407. We can see that 407 is the lowest price and matches the assignment the Hungarian algorithm determined. \(_\square\)

The Hungarian algorithm can also be executed by manipulating the weights of the bipartite graph in order to find a stable, maximum (or minimum) weight matching. This can be done by finding a feasible labeling of a graph that is perfectly matched, where a perfect matching is denoted as every vertex having exactly one edge of the matching.

How do we know that this creates a maximum-weight matching?

A feasible labeling on a perfect match returns a maximum-weighted matching. Suppose each edge \(e\) in the graph \(G\) connects two vertices, and every vertex \(v\) is covered exactly once. With this, we have the following inequality: \[w(M’) = \sum_{e\ \epsilon\ E} w(e) \leq \sum_{e\ \epsilon\ E } \big(l(e_x) + l(e_y)\big) = \sum_{v\ \epsilon\ V} l(v),\] where \(M’\) is any perfect matching in \(G\) created by a random assignment of vertices, and \(l(x)\) is a numeric label to node \(x\). This means that \(\sum_{v\ \epsilon\ V}\ l(v)\) is an upper bound on the cost of any perfect matching. Now let \(M\) be a perfect match in \(G\), then \[w(M) = \sum_{e\ \epsilon\ E} w(e) = \sum_{v\ \epsilon\ V}\ l(v).\] So \(w(M’) \leq w(M)\) and \(M\) is optimal. \(_\square\)

Start the algorithm by assigning any weight to each individual node in order to form a feasible labeling of the graph \(G\). This labeling will be improved upon by finding augmenting paths for the assignment until the optimal one is found.

A feasible labeling is a labeling such that

\(l(x) + l(y) \geq w(x,y)\ \forall x \in X, y \in Y\), where \(X\) is the set of nodes on one side of the bipartite graph, \(Y\) is the other set of nodes, \(l(x)\) is the label of \(x\), etc., and \(w(x,y)\) is the weight of the edge between \(x\) and \(y\).

A simple feasible labeling is just to label a node with the number of the largest weight from an edge going into the node. This is certain to be a feasible labeling because if \(A\) is a node connected to \(B\), the label of \(A\) plus the label of \(B\) is greater than or equal to the weight \(w(x,y)\) for all \(y\) and \(x\).

A feasible labeling of nodes, where labels are in red [2] .

Imagine there are four soccer players and each can play a few positions in the field. The team manager has quantified their skill level playing each position to make assignments easier.

How can players be assigned to positions in order to maximize the amount of skill points they provide?

The algorithm starts by labeling all nodes on one side of the graph with the maximum weight. This can be done by finding the maximum-weighted edge and labeling the adjacent node with it. Additionally, match the graph with those edges. If a node has two maximum edges, don’t connect them.

Although Eva is the best suited to play defense, she can't play defense and mid at the same time!

If the matching is perfect, the algorithm is done as there is a perfect matching of maximum weights. Otherwise, there will be two nodes that are not connected to any other node, like Tom and Defense. If this is the case, begin iterating.

Improve the labeling by finding the non-zero label vertex without a match, and try to find the best assignment for it. Formally, the Hungarian matching algorithm can be executed as defined below:

The Hungarian Algorithm for Graphs [3] Given: the labeling \(l\), an equality graph \(G_l = (V, E_l)\), an initial matching \(M\) in \(G_l\), and an unmatched vertex \(u \in V\) and \(u \notin M\) Augmenting the matching A path is augmenting for \(M\) in \(G_l\) if it alternates between edges in the matching and edges not in the matching, and the first and last vertices are free vertices , or unmatched, in \(M\). We will keep track of a candidate augmenting path starting at the vertex \(u\). If the algorithm finds an unmatched vertex \(v\), add on to the existing augmenting path \(p\) by adding the \(u\) to \(v\) segment. Flip the matching by replacing the edges in \(M\) with the edges in the augmenting path that are not in \(M\) \((\)in other words, the edges in \(E_l - M).\) Improving the labeling \(S \subseteq X\) and \(T \subseteq Y,\) where \(S\) and \(T\) represent the candidate augmenting alternating path between the matching and the edges not in the matching. Let \(N_l(S)\) be the neighbors to each node that is in \(S\) along edges in \(E_l\) such that \(N_l(S) = \{v|\forall u \in S: (u,v) \in E_l\}\). If \(N_l(S) = T\), then we cannot increase the size of the alternating path (and therefore can't further augment), so we need to improve the labeling. Let \(\delta_l\) be the minimum of \(l(u) + l(v) - w(u,v)\) over all of the \(u \in S\) and \(v \notin T\). Improve the labeling \(l\) to \(l'\): If \(r \in S,\) then \(l'(r) = l(r) - \delta_l,\) If \(r \in T,\) then \(l'(r) = l(r) + \delta_l.\) If \(r \notin S\) and \(r \notin T,\) then \(l'(r) = l(r).\) \(l'\) is a valid labeling and \(E_l \subset E_{l'}.\) Putting it all together: The Hungarian Algorithm Start with some matching \(M\), a valid labeling \(l\), where \(l\) is defined as the labelling \(\forall x \in X, y \in Y| l(y) = 0, l(x) = \text{ max}_{y \in Y}(w\big(x, y)\big)\). Do these steps until a perfect matching is found \((\)when \(M\) is perfect\():\) (a) Look for an augmenting path in \(M.\) (b) If an augmenting path does not exist, improve the labeling and then go back to step (a).

Each step will increase the size of the matching \(M\) or it will increase the size of the set of labeled edges, \(E_l\). This means that the process will eventually terminate since there are only so many edges in the graph \(G\). [4]

When the process terminates, \(M\) will be a perfect matching. By the Kuhn-Munkres theorem , this means that the matching is a maximum-weight matching.

The algorithm defined above can be implemented in the soccer scenario. First, the conflicting node is identified, implying that there is an alternating tree that must be reconfigured.

There is an alternating path between defense, Eva, mid, and Tom.

To find the best appropriate node, find the minimum \(\delta_l\), as defined in step 4 above, where \(l_u\) is the label for player \(u,\) \(l_v\) is the label for position \(v,\) and \(w_{u, v}\) is the weight on that edge.

The \(\delta_l\) of each unmatched node is computed, where the minimum is found to be a value of 2, between Tom playing mid \((8 + 0 – 6 = 2).\)

The labels are then augmented and the new edges are graphed in the example. Notice that defense and mid went down by 2 points, whereas Eva’s skillset got back two points. However, this is expected as Eva can't play in both positions at once.

Augmenting path leads to relabeling of nodes, which gives rise to the maximum-weighted path.

These new edges complete the perfect matching of the graph, which implies that a maximum-weighted graph has been found and the algorithm can terminate.

The complexity of the algorithm will be analyzed using the graph-based technique as a reference, yet the result is the same as for the matrix-based one.

Algorithm analysis [3] At each \(a\) or \(b\) step, the algorithm adds one edge to the matching and this happens \(O\big(|V|\big)\) times. It takes \(O\big(|V|\big)\) time to find the right vertex for the augmenting (if there is one at all), and it is \(O\big(|V|\big)\) time to flip the matching. Improving the labeling takes \(O\big(|V|\big)\) time to find \(\delta_l\) and to update the labelling accordingly. We might have to improve the labeling up to \(O\big(|V|\big)\) times if there is no augmenting path. This makes for a total of \(O\big(|V|^2\big)\) time. In all, there are \(O\big(|V|\big)\) iterations each taking \(O\big(|V|\big)\) work, leading to a total running time of \(O\big(|V|^3\big)\).

• Matching Algorithms
• Bruff, D. The Assignment Problem and the Hungarian Method . Retrieved June 26, 2016, from http://www.math.harvard.edu/archive/20_spring_05/handouts/assignment_overheads.pdf
• Golin, M. Bipartite Matching & the Hungarian Method . Retrieved Retrieved June 26th, 2016, from http://www.cse.ust.hk/~golin/COMP572/Notes/Matching.pdf
• Grinman, A. The Hungarian Algorithm for Weighted Bipartite Graphs . Retrieved June 26, 2016, from http://math.mit.edu/~rpeng/18434/hungarianAlgorithm.pdf
• Golin, M. Bipartite Matching & the Hungarian Method . Retrieved June 26, 2016, from http://www.cse.ust.hk/~golin/COMP572/Notes/Matching.pdf

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Assignment as a Minimum Cost Flow Problem

You can use the min cost flow solver to solve special cases of the assignment problem .

In fact, min cost flow can often return a solution faster than either the MIP or CP-SAT solver. However, MIP and CP-SAT can solve a larger class of problems than min cost flow, so in most cases MIP or CP-SAT are the best choices.

The following sections present Python programs that solve the following assignment problems using the min cost flow solver:

• A minimal linear assignment example .
• An assignment problem with teams of workers .

Linear assignment example

This section show how to solve the example, described in the section Linear Assignment Solver , as a min cost flow problem.

Import the libraries

The following code imports the required library.

Declare the solver

The following code creates the minimum cost flow solver.

Create the data

The flow diagram for the problem consists of the bipartite graph for the cost matrix (see the assignment overview for a slightly different example), with a source and sink added.

The data contains the following four arrays, corresponding to the start nodes, end nodes, capacities, and costs for the problem. The length of each array is the number of arcs in the graph.

To make clear how the data is set up, each array is divided into three sub-arrays:

• The first array corresponds to arcs leading out of the source.
• The second array corresponds to the arcs between workers and tasks. For the costs , this is just the cost matrix (used by the linear assignment solver), flattened into a vector.
• The third array corresponds to the arcs leading into the sink.

The data also includes the vector supplies , which gives the supply at each node.

How a min cost flow problem represents an assignment problem

How does the min cost flow problem above represent an assignment problem? First, since the capacity of every arc is 1, the supply of 4 at the source forces each of the four arcs leading into the workers to have a flow of 1.

Next, the flow-in-equals-flow-out condition forces the flow out of each worker to be 1. If possible, the solver would direct that flow across the minimum cost arc leading out of each worker. However, the solver cannot direct the flows from two different workers to a single task. If it did, there would be a combined flow of 2 at that task, which couldn't be sent across the single arc with capacity 1 from the task to the sink. This means that the solver can only assign a task to a single worker, as required by the assignment problem.

Finally, the flow-in-equals-flow-out condition forces each task to have an outflow of 1, so each task is performed by some worker.

Create the graph and constraints

The following code creates the graph and constraints.

Invoke the solver

The following code invokes the solver and displays the solution.

The solution consists of the arcs between workers and tasks that are assigned a flow of 1 by the solver. (Arcs connected to the source or sink are not part of the solution.)

The program checks each arc to see if it has flow 1, and if so, prints the Tail (start node) and the Head (end node) of the arc, which correspond to a worker and task in the assignment.

Output of the program

Here is the output of the program.

The result is the same as that for the linear assignment solver (except for the different numbering of workers and costs). The linear assignment solver is slightly faster than min cost flow — 0.000147 seconds versus 0.000458 seconds.

The entire program

The entire program is shown below.

Assignment with teams of workers

This section presents a more general assignment problem. In this problem, six workers are divided into two teams. The problem is to assign four tasks to the workers so that the workload is equally balanced between the teams — that is, so each team performs two of the tasks.

For a MIP solver solution to this problem see Assignment with Teams of Workers .

The following sections describe a program that solves the problem using the min cost flow solver.

The following code creates the data for the program.

The workers correspond to nodes 1 - 6. Team A consists of workers 1, 3, and 5, and team B consists of workers 2, 4, and 6. The tasks are numbered 7 - 10.

There are two new nodes, 11 and 12, between the source and workers. Node 11 is connected to the nodes for team A, and Node 12 is connected to the nodes for team B, with arcs of capacity 1. The graph below shows just the nodes and arcs from the source to the workers.

The key to balancing the workload is that the source 0 is connected to nodes 11 and 12 by arcs of capacity 2. This means that nodes 11 and 12 (and therefore teams A and B) can have a maximum flow of 2. As a result, each team can perform at most two of the tasks.

Create the constraints

The following shows the output of the program.

Team A is assigned tasks 9 and 10, while team B is assigned tasks 7 and 8.

Note that the min cost flow solver is faster for this problem than the MIP solver , which takes around 0.006 seconds.

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 2023-09-21 UTC.

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• For each row of the matrix, find the smallest element and subtract it from every element in its row.
• Do the same (as step 1) for all columns.
• Cover all zeros in the matrix using minimum number of horizontal and vertical lines.
• Test for Optimality: If the minimum number of covering lines is n, an optimal assignment is possible and we are finished. Else if lines are lesser than n, we haven’t found the optimal assignment, and must proceed to step 5.
• Determine the smallest entry not covered by any line. Subtract this entry from each uncovered row, and then add it to each covered column. Return to step 3.

Try it before moving to see the solution

Explanation for above simple example:

  An example that doesn’t lead to optimal value in first attempt: In the above example, the first check for optimality did give us solution. What if we the number covering lines is less than n.

Time complexity : O(n^3), where n is the number of workers and jobs. This is because the algorithm implements the Hungarian algorithm, which is known to have a time complexity of O(n^3).

Space complexity :   O(n^2), where n is the number of workers and jobs. This is because the algorithm uses a 2D cost matrix of size n x n to store the costs of assigning each worker to a job, and additional arrays of size n to store the labels, matches, and auxiliary information needed for the algorithm.

In the next post, we will be discussing implementation of the above algorithm. The implementation requires more steps as we need to find minimum number of lines to cover all 0’s using a program. References: http://www.math.harvard.edu/archive/20_spring_05/handouts/assignment_overheads.pdf https://www.youtube.com/watch?v=dQDZNHwuuOY

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Procedure, Example Solved Problem | Operations Research - Solution of assignment problems (Hungarian Method) | 12th Business Maths and Statistics : Chapter 10 : Operations Research

Chapter: 12th business maths and statistics : chapter 10 : operations research.

Solution of assignment problems (Hungarian Method)

First check whether the number of rows is equal to the numbers of columns, if it is so, the assignment problem is said to be balanced.

Step :1 Choose the least element in each row and subtract it from all the elements of that row.

Step :2 Choose the least element in each column and subtract it from all the elements of that column. Step 2 has to be performed from the table obtained in step 1.

Step:3 Check whether there is atleast one zero in each row and each column and make an assignment as follows.

Step :4 If each row and each column contains exactly one assignment, then the solution is optimal.

Example 10.7

Solve the following assignment problem. Cell values represent cost of assigning job A, B, C and D to the machines I, II, III and IV.

Here the number of rows and columns are equal.

∴ The given assignment problem is balanced. Now let us find the solution.

Step 1: Select a smallest element in each row and subtract this from all the elements in its row.

Look for atleast one zero in each row and each column.Otherwise go to step 2.

Step 2: Select the smallest element in each column and subtract this from all the elements in its column.

Since each row and column contains atleast one zero, assignments can be made.

Step 3 (Assignment):

Thus all the four assignments have been made. The optimal assignment schedule and total cost is

The optimal assignment (minimum) cost

Example 10.8

Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows. Determine the optimum assignment schedule.

∴ The given assignment problem is balanced.

Now let us find the solution.

The cost matrix of the given assignment problem is

Column 3 contains no zero. Go to Step 2.

Thus all the five assignments have been made. The Optimal assignment schedule and total cost is

The optimal assignment (minimum) cost = ` 9

Example 10.9

Solve the following assignment problem.

Since the number of columns is less than the number of rows, given assignment problem is unbalanced one. To balance it , introduce a dummy column with all the entries zero. The revised assignment problem is

Here only 3 tasks can be assigned to 3 men.

Step 1: is not necessary, since each row contains zero entry. Go to Step 2.

Step 3 (Assignment) :

Since each row and each columncontains exactly one assignment,all the three men have been assigned a task. But task S is not assigned to any Man. The optimal assignment schedule and total cost is

The optimal assignment (minimum) cost = ₹ 35

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scipy.optimize.linear_sum_assignment ¶

Solve the linear sum assignment problem.

The linear sum assignment problem is also known as minimum weight matching in bipartite graphs. A problem instance is described by a matrix C, where each C[i,j] is the cost of matching vertex i of the first partite set (a “worker”) and vertex j of the second set (a “job”). The goal is to find a complete assignment of workers to jobs of minimal cost.

Formally, let X be a boolean matrix where \(X[i,j] = 1\) iff row i is assigned to column j. Then the optimal assignment has cost

s.t. each row is assignment to at most one column, and each column to at most one row.

This function can also solve a generalization of the classic assignment problem where the cost matrix is rectangular. If it has more rows than columns, then not every row needs to be assigned to a column, and vice versa.

The method used is the Hungarian algorithm, also known as the Munkres or Kuhn-Munkres algorithm.

New in version 0.17.0.

• http://csclab.murraystate.edu/bob.pilgrim/445/munkres.html
• Harold W. Kuhn. The Hungarian Method for the assignment problem. Naval Research Logistics Quarterly , 2:83-97, 1955.
• Harold W. Kuhn. Variants of the Hungarian method for assignment problems. Naval Research Logistics Quarterly , 3: 253-258, 1956.
• Munkres, J. Algorithms for the Assignment and Transportation Problems. J. SIAM , 5(1):32-38, March, 1957.
• https://en.wikipedia.org/wiki/Hungarian_algorithm

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Mask2Former - ValueError: cost matrix is infeasible #21644

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How to Make a Responsibility Assignment Matrix: Excel RACI Template

What is a responsibility assignment matrix?

How to create a responsibility assignment matrix in excel, free raci template for excel, how to manage raci roles in your teamgantt plan.

A responsibility assignment matrix (RAM) is a tool used in project management to clarify team and stakeholder roles for each project step. It paves the way for smooth collaboration by ensuring everyone knows what they need to do, who they need to talk to, and who has the final say on key decisions and deliverables.

RACI—which stands for Responsible , Accountable , Consulted , Informed —is the most popular framework used for assigning roles and responsibilities on projects. Here’s a quick breakdown of RACI categories in basic terms:

• Responsible : Who completes the work?
• Accountable : Who makes decisions? 
• Consulted : Who provides expertise?
• Informed : Who needs status updates?

Of course, RACI isn’t the only responsibility assignment matrix out there. These RACI alternatives provide a small sample of other approaches you might come across in project management.

• RASCI (or RASIC) matrix : This RACI alternative adds one extra role into the responsibility assignment mix. In the RASCI model, the S stands for Supportive . While this role covers anyone who will lend the Responsible person a hand with the work, a Supportive team member isn’t responsible for the outcome.
• DACI matrix : DACI stands for Driver , Approver , Contributor , Informed and is used to outline decision-making roles and responsibilities for projects. In this framework, the project manager or leader typically serves as the Driver guiding the team to a decision.‍
• RAPID responsibility matrix : RAPID stands for Recommend , Agree , Perform , Input , Decide and is another decision-making framework used to define authority vs accountability. The Recommend role kicks things off by suggesting an action, while the Decide role has the ultimate say in how things move forward. ‍
• CARS : CARS stands for Communicate , Approver , Responsible , Support . In this model, Communicate combines RACI’s Consulted and Informed roles into a single assignment. Someone with the Communicate role lends their expertise and needs to be kept up-to-date on progress. The Approver is the main decision-maker who calls the shots.

Lots of people use spreadsheets to make a responsibility assignment matrix for their projects, so let’s walk through the basic steps of building one in Excel, using the RACI framework as our model.

Looking for an online solution? See how TeamGantt's RACI feature integrates into your project plan.

1. List project tasks and deliverables in column A

First, make a list of all the work that needs to be done for your project down the left side of your matrix. Enter each project task, milestone, or decision in column A of your Excel worksheet. 

Feel free to group tasks by project phase like we’ve done in the screenshot below. That way, your RACI matrix is easy to scan and read.

2. Add team members or project roles across row 1

Starting with column B, label each column header with the name of a team member and/or project role. 

Include the people who will execute and review work for the project, as well as any subject matter experts or stakeholders you may need to consult or keep in the loop along the way.

3. Insert a new worksheet for roles and definitions

Click Insert > Insert Sheet from the Home ribbon at the top of your Excel workbook.

Go to your new worksheet, and list each letter of the RACI acronym in column A. Then enter the corresponding role for each letter in column B. We also included RACI definitions in column C as a handy reference for anyone who might need a refresher.

You’ll use this worksheet to populate a drop-down list on the main RACI matrix tab to make it easier to assign roles quickly.

4. Add a drop-down list of roles to your matrix

Now, go back to your main worksheet, and click into the first open cell in your matrix.

On the ribbon, click Data > Data validation to insert a drop-down list with RACI roles.

On the Settings tab, choose List under the Allow menu.

Click into the Source field, then highlight the data range with your options from the RACI Roles & Definitions worksheet you set up in Step 3. We highlighted cells A2-A5 in our example.

Verify your Data validation settings are correct, then hit Enter to add the drop-down list to your selected cell.

Copy and paste that cell to apply the drop-down list to other cells in your RACI matrix worksheet.

5. Color-code assignments with conditional formatting

Click Conditional Formatting > New Rule on the Home tab. Select Classic > Format only cells that contain > Specific text > containing . Enter the letter R in the text box, then choose Custom Format , and apply a background color (and any other styles you want). 

Repeat this step for each additional letter in the acronym.

6. Assign a RACI value to everyone on every task

You’re almost there! Now go down the list of tasks on your responsibility assignment matrix, and assign a role to every person who will be involved in that project step or deliverable.

Want to build a responsibility assignment matrix of your own, but don't want to start from scratch? Download our ready-made Excel template for free. This blank RACI template is fully editable, so you can customize it for any project you manage. 

We added drop-downs for assigning RACI roles more easily and included a RACI chart example tab as reference in case you need a little extra guidance.

Download: RACI matrix template for Excel

You can easily upload your final matrix to your TeamGantt project . But if you don’t want to worry about outdated spreadsheets that get forgotten once work begins, why not assign RACI roles directly to your plan?

Here’s how to use TeamGantt’s online RACI feature for your next project.

Assigning RACI roles and responsibilities to TeamGantt tasks

• Open your project, and toggle to the RACI tab. This will display all your project tasks in a list format (rows). On the right side of the matrix, you’ll see a column for each person currently invited to the project with cells for each task in the project. 
• Click the cell below each person who needs to be assigned a role on a task, and choose one of the RACI options from the drop-down.

Viewing RACI matrix assignments for your project

There are 2 simple ways to view RACI assignments in TeamGantt:

• From the Gantt tab : If someone is assigned to a task and has a RACI role on that task, the RACI value will appear in parentheses next to that person’s name on the gantt chart. Just be aware that you won’t see RACI assignments for people who haven’t been assigned to a specific task in Gantt view.

• From the RACI tab : To access your project’s full RACI matrix, simply toggle to the RACI tab for that project. You’ll find RACI assignments for every person playing a role—whether or not they’re the one responsible for doing the work.

Keep teams in sync—and accountable—with TeamGantt

A RACI chart is a simple tool that makes projects easier to manage by creating less confusion and more accountability. But you’ve got more than roles and responsibilities to keep straight.

TeamGantt makes it easy to build a project plan your whole team can contribute to and collaborate on. Everything happens online, so you can stay on top of deadlines and monitor progress in real time.

Use our built-in RACI chart to assign roles and keep them visible from project start to finish, so everyone knows how they contribute to success.

Try TeamGantt’s Pro Manager plan free for 30 days!

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Key takeaways

Successful project management depends on a team-wide understanding of roles and responsibilities. Using a RACI matrix to assign and define each role is a great way to keep a project on track and positioned for success.

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How Does a RACI Chart Help Project Managers?

Project managers use RACI charts to keep track of team roles and relay those responsibilities to the larger team. The matrix defines clear roles and responsibilities for individual team members across the various phases of the project, breaking each role down into four types of designation: those who are Responsible and Accountable for project deliverables, those who should be Consulted as work begins, and stakeholders who need to be Informed of ongoing progress, roadblocks, and updates. 

Read more: Project Management Phases

RACI Matrix Definitions 

Responsible.

The individual(s) with responsibility for the task or deliverable is typically responsible for developing and completing the project deliverables themselves. The responsible parties are typically hands-on team members who make direct contributions toward the completion of the project. The responsible team is comprised of the project’s “doers”, working hands-on to ensure that each deliverable is completed. 

Some examples of responsible parties are:

• Project Managers
• Business Analysts
• Graphic Designers
• Copywriters

Accountable

Accountable parties ensure accountability to project deadlines, and ultimately, accountability to project completion. This group frequently also falls under the informed category.

Some examples of accountable parties are:

• Product Owners
• Signature Authorities
• Business Owners
• Key Stakeholders

Consulted individuals’ opinions are crucial, and their feedback needs to be considered at every step of the game. These individuals provide guidance that is often a prerequisite to other project tasks, for example, providing legal guidance on a project throughout the process. If you are working on new product development or expansion, this could essentially be the entire organization.

Some examples of consulted parties are:

• Legal Experts
• Information Security and Cybersecurity Experts
• Compliance Consultants

Informed persons are those that need to stay in the loop of communication throughout the project. These individuals do not have to be consulted or be a part of the decision-making, but they should be made aware of all project updates. Typically, this party are business owners or stakeholders that are more interested in viewing the project at a 30,000-foot view.  Keep this group on your cc list for awareness of topics, decisions, and progress – that includes making them part of the initial project kickoff and project demos as optional attendees. This group often also falls under the accountable group.

Some examples of informed parties are:

• Project Committee Members
• External Stakeholders

Read more: DACI vs RACI Model Guide

Why Are RACI Roles Important?

RACI roles provide a sense of organization and clarity for teams that are looking to divide roles and keep team members accountable for their contributions. Considering that 27% of projects go over budget, for reasons like scope creep and lack of defined roles, RACI roles help position a project for success and avoid common pitfalls. 

Moreover, RACI roles help ensure that communication between all roles is ongoing. When you consider that nearly half of all project spending is at risk of being wasted due to a lack of effective team-based communication, it becomes all that more important to prioritize. Ultimately, teams who prioritize communication and well-defined roles are better off, and RACI roles help teams achieve that goal faster – while providing accountability for each team member’s unique contributions to the success of the project. 

Read More: Top 10 Main Causes of Project Failure

How to Create a RACI Matrix 

If you’re looking to implement a RACI matrix as part of your team’s project planning process, take these steps to create a RACI matrix.

Ensure that you have a thorough understanding of the project and its demands before outlining any further steps by communicating with key stakeholders and decision-makers.

Determine the list of key activities and deliverables from the director of program management or other leadership. 

Determine who is needed to be a part of the project or initiative.

Determine the project roles and responsible job titles and persons for each activity and deliverable.

Hold review sessions with key members of the team for alignment, and if you haven’t already, host a kickoff meeting with the entirety of the team and key stakeholders to unveil the matrix, address questions, and more. 

If the project has already started, it’s not too late to implement a RACI matrix.

• Outline the story. Using research from multiple sources, do a, b, c, and d.
• Utilize steps 2 and 3 (shown above). Ensure the right groups are assigned and engaged. 
• Hold a review session. Ensure that the team acknowledges and discusses the plan and the roles assigned.

Read more: 8 Factors That Lead to Successful Projec ts

Examples of a RACI Matrix

As shown above, a RACI matrix helps break down what roles individuals will play as work is carried out and to what extent they will be involved in the project overall. The horizontal axis represents each person on the project team and the vertical axis represents each task.

Each square of the matrix represents an individual, a task, and that individual’s role within the project, either responsible, accountable, consulted, or informed. In this situation, for example, the project manager is accountable for accessing risk, defining performance requirements, creating designs, executing construction, and approving construction work. However, they are only informed about approving construction work and defining functional and aesthetic needs.

Read more: Understanding Different Types of Stakeholders and Their Roles

Our FREE Downloadable RACI Matrix Template

Who creates the raci matrix.

The RACI matrix — sometimes called RACI model, RACI diagram, or simply just RAC — is created by the project manager at the start of the project as a key part of establishing the initial human resources planning for the project. Because miscommunication is a common threat to any project, RACI charts are a great asset to teams dealing with any type of project, from very simple projects to extremely complex ones. 

Common Mistakes When Creating a RACI Matrix

• Failure to plan ahead: Utilizing a RACI matrix should not be your first step in project planning. Having a fully assembled project team and at least a general idea of a task list and project plans is a better place to start before preparing a matrix.
• Working with too large a team: A RACI matrix is likely not the best bet for a large team, as it will make the matrix hard to understand and overly complex.
• Not communicating with the project team: A RACI matrix should help organize tasks and responsibilities that have already been introduced to the project team – no one likes to be blindsided. Be sure to host a kickoff meeting with the team first before creating a RACI matrix.

Frequently Asked Questions

Implementing a RACI matrix takes more than just a few emails and sporadic conversations – it takes consistent communication and planning. You should host a kickoff meeting to introduce the matrix to the team and make a plan to continue meeting at predetermined times throughout the project lifecycle. 

Here are a few more tips to keep in mind as you implement your RACI matrix within the team dynamic:

• Get everyone prepared. Send the document around to the meeting distribution as read-ahead material, requesting feedback if there are any major concerns. 
• Roll out each role for the team . During the meeting, conduct a review of the tasks and responsible parties. Do not rush through this review, but rather ensure enough time in your project kickoff for this important aspect. (Be certain to clarify the definitions of RACI to avoid ambiguity.)
• Consider changes and update accordingly. After the meeting, send out the notes documenting acceptance or updates to the RACI. In addition to sending out the notes, request any corrections within a reasonable yet defined timeframe. Clarify that if no changes are requested, each person is acknowledging their role and committing to the project tasks as outlined.
• Stay in touch. Consider a quick review with the entire team each quarter or every six months for longer projects to ensure it remains up-to-date and not simply another document in the repository but a relied-upon artifact.

As you implement the RACI matrix…

• Encourage teamwork and foster collaboration whenever possible.
• Don’t fear updates – make changes and adjustments as needed (but be sure to communicate those changes clearly to all parties).
• Earlier is better. Roll out your matrix plan to the team BEFORE you plan to implement it for the best results. 
• Have a clear-cut understanding of the project scope and how each role connects to the overall project goal.

For “Responsible” Parties:

• Make sure your project’s definition of Responsible is clear on who holds the “decider” role for the project or project phase’s completion, and what the dimensions of that responsibility will be.
• Ensure that all parties are aware of their role and responsibilities within the matrix.

For “Accountable” Parties: 

• When multiple Accountable team members must exist, use your definitions to make clear which individual is accountable for a given project element, and how that individual needs to interact with other Accountable team members.
• Ensure that there is only one “Accountable” party assigned per task.
• Be sure that the Accountable party has the authority and power to oversee the task as the accountable party.

For Consulted and Informed Parties: 

• Consulted parties are often high-level decision-makers with heavy schedules. Make sure you’re clear on their availability ahead of time.
• Similar to Consulted parties, Informed parties are often less hands-on and have less understanding of day-to-day project operations. As the project goes on, make sure to keep detailed notes to keep the Informed party up-to-date on key information. 
• Understand the ways that these parties like to communicate and create a plan to reach them early – whether that’s over phone calls, emails, video calls, or from within your project management system’s collaboration tools.
• Knowing the difference between who needs to be consulted versus informed can be a challenge if there is ambiguity about project roles. Consider what aspects of the project different team members need to know to do their jobs, and then bake those into your definitions.

RACI Matrix Pros & Cons

• Increased Engagement: RACI helps engage project participants in the project lifecycle. 
• Enhanced Project Planning: Project managers make project planning more organized, efficient, and detailed.
• Identifiable Improvement Opportunities: Areas of improvement are more easily identified.
• Easier Collaboration: Use of a RACI matrix creates a clear path for leadership to sign off on project steps, as project documentation in the RACI model is heavily emphasized.
• Better Communication: Improves overall group communication as a whole.
• Group Accountability: Assists groups, especially larger project teams, stay connected and accountable to their roles and project goals
• Limitations on Role Scope: The RACI model does not provide details on role scope, especially for responsible parties. These gaps in detail also affect other team roles, for example, another gap in a RACI is the determination of who is responsible for verifier and signatory.
• Limits on Task Details and Scope: While a RACI matrix can provide an overview of who is responsible for different tasks, it will not state what needs to be done.
• Not Aligned to the Agile Methodology: Project managers using an agile methodology like scrum may find it redundant since accountability, ownership, and ongoing communication is built into the scrum framework (i.e., product owner, scrum master, and daily standups with the team). Additionally, agile focuses on team-based delivery and accountability, while the RACI framework and alternatives focus on individual responsibility and autonomous accountability.

Read more: Top 10 Causes of Project Failure

Free RACI Matrix Templates

A number of project management software solutions include a native RACI matrix template. Here are just a few we’ve found:

Colorful RACI Chart Template

We love this template from Smartsheet because it’s colorful, thorough, and includes room for every party involved in the project. 

Pastel Colored RACI Matrix Template

This template from the Academy to Innovate HR is a great choice for project managers who want to organize their team roles with an easy-on-the-eyes chart that evolves beyond the simple spreadsheet. 

Simple RACI Chart from Clickup

These RACI templates from Clickup have enough variety to fit any of your project needs, but are simple enough for even beginner PMs to use.

Detailed RACI Matrix Template

This template is a great starter template for anyone looking to explore RACI charts in their project management strategy . As an added bonus – it comes with the RACI definitions already built in!

Excel-Based RACI Chart Template

Are you an Excel or Google Sheets user looking to take advantage of the RACI matrix? An Excel-formatted template from Project Management Docs can be just the solution for you. This template is a great template for users who want a chart that comes in a pre-formatted structure.

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Watch CBS News

How airline "drip pricing" can disguise the true cost of flying

By Megan Cerullo

Edited By Alain Sherter

Updated on: April 24, 2024 / 2:47 PM EDT / CBS News

With many airlines now hawking "unbundled" fares, it's easy for travelers to mistake low advertised prices for cheap plane tickets . But for consumers eager to get the best deal on flights heading into the summer travel season, it pays to learn how "drip pricing" can make airfare more expensive.

Indeed, selecting the cheapest base fare is no longer the best way to get a good deal, according to travel experts. That's because airlines now routinely charge more money for "extras" such as seat assignments,  checked bags , snacks or wifi. 

"Nobody likes feeling nickel-and-dimed, like the price they saw for a flight was a bait and switch," Scott Keyes, founder and CEO of travel site Going.com, told CBS MoneyWatch.

Here's what to consider. At first glance, the initial pricing for a flight you find on an online travel site might seem temptingly low. But after factoring in the cost of selecting your seat, checking bags and other add-ons, the fare can end up being much higher — as much or more than an all-inclusive fare.

This model, commonly referred to as drip pricing, can certainly boost an airline's revenue, and proponents say it benefits consumers by allowing them to pay only for the perks they truly value. For their part, critics say it makes it harder to determine the true cost of flying and to compare prices among airlines.

Keyes traces drip pricing back to 2008, when airlines began charging passengers to check second bags. That allowed full-service carriers to offer a lower-cost, no-frills ticket in order to compete with budget carriers.

"That lower headline price brought people in — then they started adding seat-selection fees," Keyes said. "It's an innovation from the budget airlines that the entire industry has copied and that full-service airlines have adopted for themselves."

"It makes it very difficult"

For consumers, however, the problem with unbundling fares is it makes it trickier to compare what different airlines charge for tickets, experts told CBS MoneyWatch. 

"It makes it very difficult to find out what the all-in price will be," said Columbia Business School marketing professor Vicki Morwitz, who authored a  report on how consumers react to drip pricing.

Her research shows that consumers tend to book the ticket option that looks cheaper upfront, but costs more once add-ons are factored in.  "Consumers make a mistake and spend more money than they needed to spend," she explained. 

Jay Sorensen, president of IdeaWorks, a consultancy that has advised U.S. airlines, agrees that drip pricing makes comparing airline ticket prices more complicated. But he still thinks it can benefit consumers by letting them pay for the extras they want, while leaving behind those that aren't important to them. 

"The outcome is of course that it's more difficult to compare between different products and airlines," he said. "While that's true, airlines, as profit-seeking companies, are under no obligation to make it easier to compare with their competitors."

Sorensen compared the experience of booking airfare today to shopping for groceries.

"You roll in with your shopping cart, and as you walk through the aisles you toss stuff in your cart," he said. "You buy a base fare, and as you go through the booking path you add things to the cart, like a checked bag, seat assignment, or pay to book a meal or other services," he said. "That's dramatically different from the way travel was once sold in U.S."

Megan Cerullo is a New York-based reporter for CBS MoneyWatch covering small business, workplace, health care, consumer spending and personal finance topics. She regularly appears on CBS News 24/7 to discuss her reporting.

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