assignment matrix is always a

Assignment Problem: Meaning, Methods and Variations | Operations Research

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Definition of Assignment Problem:

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Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:

How to Solve the Assignment Problem: A Complete Guide

Table of Contents

Assignment problem is a special type of linear programming problem that deals with assigning a number of resources to an equal number of tasks in the most efficient way. The goal is to minimize the total cost of assignments while ensuring that each task is assigned to only one resource and each resource is assigned to only one task. In this blog, we will discuss the solution of the assignment problem using the Hungarian method, which is a popular algorithm for solving the problem.

Understanding the Assignment Problem

Before we dive into the solution, it is important to understand the problem itself. In the assignment problem, we have a matrix of costs, where each row represents a resource and each column represents a task. The objective is to assign each resource to a task in such a way that the total cost of assignments is minimized. However, there are certain constraints that need to be satisfied – each resource can be assigned to only one task and each task can be assigned to only one resource.

Solving the Assignment Problem

There are various methods for solving the assignment problem, including the Hungarian method, the brute force method, and the auction algorithm. Here, we will focus on the steps involved in solving the assignment problem using the Hungarian method, which is the most commonly used and efficient method.

Step 1: Set up the cost matrix

The first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively.

Step 2: Subtract the smallest element from each row and column

To simplify the calculations, we need to reduce the size of the cost matrix by subtracting the smallest element from each row and column. This step is called matrix reduction.

Step 3: Cover all zeros with the minimum number of lines

The next step is to cover all zeros in the matrix with the minimum number of horizontal and vertical lines. This step is called matrix covering.

Step 4: Test for optimality and adjust the matrix

To test for optimality, we need to calculate the minimum number of lines required to cover all zeros in the matrix. If the number of lines equals the number of rows or columns, the solution is optimal. If not, we need to adjust the matrix and repeat steps 3 and 4 until we get an optimal solution.

Step 5: Assign the tasks to the agents

The final step is to assign the tasks to the agents based on the optimal solution obtained in step 4. This will give us the most cost-effective or profit-maximizing assignment.

Solution of the Assignment Problem using the Hungarian Method

The Hungarian method is an algorithm that uses a step-by-step approach to find the optimal assignment. The algorithm consists of the following steps:

• Subtract the smallest entry in each row from all the entries of the row.
• Subtract the smallest entry in each column from all the entries of the column.
• Draw the minimum number of lines to cover all zeros in the matrix. If the number of lines drawn is equal to the number of rows, we have an optimal solution. If not, go to step 4.
• Determine the smallest entry not covered by any line. Subtract it from all uncovered entries and add it to all entries covered by two lines. Go to step 3.

The above steps are repeated until an optimal solution is obtained. The optimal solution will have all zeros covered by the minimum number of lines. The assignments can be made by selecting the rows and columns with a single zero in the final matrix.

Applications of the Assignment Problem

The assignment problem has various applications in different fields, including computer science, economics, logistics, and management. In this section, we will provide some examples of how the assignment problem is used in real-life situations.

Applications in Computer Science

The assignment problem can be used in computer science to allocate resources to different tasks, such as allocating memory to processes or assigning threads to processors.

Applications in Economics

The assignment problem can be used in economics to allocate resources to different agents, such as allocating workers to jobs or assigning projects to contractors.

Applications in Logistics

The assignment problem can be used in logistics to allocate resources to different activities, such as allocating vehicles to routes or assigning warehouses to customers.

Applications in Management

The assignment problem can be used in management to allocate resources to different projects, such as allocating employees to tasks or assigning budgets to departments.

Let’s consider the following scenario: a manager needs to assign three employees to three different tasks. Each employee has different skills, and each task requires specific skills. The manager wants to minimize the total time it takes to complete all the tasks. The skills and the time required for each task are given in the table below:

The assignment problem is to determine which employee should be assigned to which task to minimize the total time required. To solve this problem, we can use the Hungarian method, which we discussed in the previous blog.

Using the Hungarian method, we first subtract the smallest entry in each row from all the entries of the row:

Next, we subtract the smallest entry in each column from all the entries of the column:

We draw the minimum number of lines to cover all the zeros in the matrix, which in this case is three:

Since the number of lines is equal to the number of rows, we have an optimal solution. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. In this case, the optimal assignments are:

• Emp 1 to Task 3
• Emp 2 to Task 2
• Emp 3 to Task 1

This assignment results in a total time of 9 units.

I hope this example helps you better understand the assignment problem and how to solve it using the Hungarian method.

Solving the assignment problem may seem daunting, but with the right approach, it can be a straightforward process. By following the steps outlined in this guide, you can confidently tackle any assignment problem that comes your way.

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Operations Research

1 Operations Research-An Overview

• History of O.R.
• Approach, Techniques and Tools
• Phases and Processes of O.R. Study
• Typical Applications of O.R
• Limitations of Operations Research
• Models in Operations Research
• O.R. in real world

2 Linear Programming: Formulation and Graphical Method

• General formulation of Linear Programming Problem
• Optimisation Models
• Basics of Graphic Method
• Important steps to draw graph
• Multiple, Unbounded Solution and Infeasible Problems
• Solving Linear Programming Graphically Using Computer
• Application of Linear Programming in Business and Industry

3 Linear Programming-Simplex Method

• Principle of Simplex Method
• Computational aspect of Simplex Method
• Simplex Method with several Decision Variables
• Two Phase and M-method
• Multiple Solution, Unbounded Solution and Infeasible Problem
• Sensitivity Analysis
• Dual Linear Programming Problem

4 Transportation Problem

• Basic Feasible Solution of a Transportation Problem
• Modified Distribution Method
• Stepping Stone Method
• Unbalanced Transportation Problem
• Degenerate Transportation Problem
• Transhipment Problem
• Maximisation in a Transportation Problem

5 Assignment Problem

• Solution of the Assignment Problem
• Unbalanced Assignment Problem
• Problem with some Infeasible Assignments
• Maximisation in an Assignment Problem
• Crew Assignment Problem

6 Application of Excel Solver to Solve LPP

• Building Excel model for solving LP: An Illustrative Example

7 Goal Programming

• Concepts of goal programming
• Goal programming model formulation
• Graphical method of goal programming
• The simplex method of goal programming
• Using Excel Solver to Solve Goal Programming Models
• Application areas of goal programming

8 Integer Programming

• Some Integer Programming Formulation Techniques
• Binary Representation of General Integer Variables
• Unimodularity
• Cutting Plane Method
• Branch and Bound Method
• Solver Solution

9 Dynamic Programming

• Dynamic Programming Methodology: An Example
• Definitions and Notations
• Dynamic Programming Applications

10 Non-Linear Programming

• Solution of a Non-linear Programming Problem
• Convex and Concave Functions
• Kuhn-Tucker Conditions for Constrained Optimisation
• Quadratic Programming
• Separable Programming
• NLP Models with Solver

11 Introduction to game theory and its Applications

• Important terms in Game Theory
• Saddle points
• Mixed strategies: Games without saddle points
• 2 x n games
• Exploiting an opponent’s mistakes

12 Monte Carlo Simulation

• Reasons for using simulation
• Monte Carlo simulation
• Limitations of simulation
• Steps in the simulation process
• Some practical applications of simulation
• Two typical examples of hand-computed simulation
• Computer simulation

13 Queueing Models

• Characteristics of a queueing model
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2.1: Introduction to Matrices

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• Page ID 37846

• Rupinder Sekhon and Roberta Bloom
• De Anza College

Learning Objectives

In this section, you will learn to:

• Add and subtract matrices.
• Multiply a matrix by a scalar.
• Multiply two matrices.

A matrix is a 2 dimensional array of numbers arranged in rows and columns. Matrices provide a method of organizing, storing, and working with mathematical information. Matrices have an abundance of applications and use in the real world. Matrices provide a useful tool for working with models based on systems of linear equations. We’ll use matrices in sections 2.2, 2.3, and 2.4 to solve systems of linear equations with several variables in this chapter.

Matrices are used in encryption, which we will explore in section 2.5 and in economic modelling, explored in section 2.6. We use matrices again in chapter 4, in optimization problems such as maximizing profit or revenue, or minimizing cost. Matrices are used in business for scheduling, routing transportation and shipments, and managing inventory.

Just about any application that collects and manages data can apply matrices. Use of matrices has grown as the availability of data in many areas of life and business has increased. They are important tools for organizing data and solving problems in all fields of science, from physics and chemistry, to biology and genetics, to meteorology, and economics. In computer science, matrix mathematics lies behind animation of images in movies and video games.

Computer science analyzes diagrams of networks to understand how things are connected to each other, such as relationships between people on a social website, and relationships between results in line search and how people link from one website to another. The mathematics to work with network diagrams comprise the field of “graph theory”; it relies on matrices to organize the information in the graphs that diagram connections and associations in a network. For example, if you use Facebook or Linked-In, or other social media sites, these sites use network graphs and matrices to organize your relationships with other users.

Introduction to Matrices

A matrix is a rectangular array of numbers. Matrices are useful in organizing and manipulating large amounts of data. In order to get some idea of what matrices are all about, we will look at the following example.

Example \(\PageIndex{1}\)

Fine Furniture Company makes chairs and tables at its San Jose, Hayward, and Oakland factories. The total production, in hundreds, from the three factories for the years 2014 and 2015 is listed in the table below.

• Represent the production for the years 2014 and 2015 as the matrices A and B.
• Find the difference in sales between the years 2014 and 2015.
• The company predicts that in the year 2020 the production at these factories will be double that of the year 2014. What will the production be for the year 2020?

a) The matrices are as follows:

\[A=\left[\begin{array}{ll} 30 & 18 \\ 20 & 12 \\ 16 & 10 \end{array}\right] \nonumber \]

\[\mathrm{B}=\left[\begin{array}{ll} 36 & 20 \\ 24 & 18 \\ 20 & 12 \end{array}\right] \nonumber \]

b) We are looking for the matrix \(B - A\). When two matrices have the same number of rows and columns, the matrices can be added or subtracted entry by entry. Therefore, we get

\[\mathrm{B}-\mathrm{A}=\left[\begin{array}{ll} 36-30 & 20-18 \\ 24-20 & 18-12 \\ 20-16 & 12-10 \end{array}\right]=\left[\begin{array}{ll} 6 & 2 \\ 4 & 6 \\ 4 & 2 \end{array}\right] \nonumber \]

c) We would like a matrix that is twice the matrix of 2014, i.e., \(2A\).

Whenever a matrix is multiplied by a number, each entry is multiplied by the number.

\[2 \mathrm{A}=2\left[\begin{array}{ll} 30 & 18 \\ 20 & 12 \\ 16 & 10 \end{array}\right]=\left[\begin{array}{ll} 60 & 36 \\ 40 & 24 \\ 32 & 20 \end{array}\right] \nonumber \]

Before we go any further, we need to familiarize ourselves with some terms that are associated with matrices. The numbers in a matrix are called the entries or the elements of a matrix.

Whenever we talk about a matrix, we need to know the size or the dimension of the matrix. The dimension of a matrix is the number of rows and columns it has. When we say a matrix is a “3 by 4 matrix”, we are saying that it has 3 rows and 4 columns. The rows are always mentioned first and the columns second. This means that a \(3 \times 4\) matrix does not have the same dimension as a \(4 \times 3\) matrix.

\[A=\left[\begin{array}{cccc} 1 & 4 & -2 & 0 \\ 3 & -1 & 7 & 9 \\ 6 & 2 & 0 & 5 \end{array}\right] \nonumber \]

\[B=\left[\begin{array}{ccc} 2 & 9 & 8 \\ -3 & 0 & 1 \\ 6 & 5 & -2 \\ -4 & 7 & 8 \end{array}\right] \nonumber \]

Matrix \(A\) has dimensions \(3 \times 4\) and matrix \(B\) has dimensions \(4 \times 3\).

A matrix that has the same number of rows as columns is called a square matrix . A matrix with all entries zero is called a zero matrix. A square matrix with 1's along the main diagonal and zeros everywhere else, is called an identity matrix . When a square matrix is multiplied by an identity matrix of same size, the matrix remains the same.

\[I=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \nonumber \]

Matrix \(I\) is a \(3 \times 3\) identity matrix

A matrix with only one row is called a row matrix or a row vector, and a matrix with only one column is called a column matrix or a column vector . Two matrices are equal if they have the same size and the corresponding entries are equal.

We can perform arithmetic operations with matrices. Next we will define and give examples illustrating the operations of matrix addition and subtraction, scalar multiplication, and matrix multiplication. Note that matrix multiplication is quite different from what you would intuitively expect, so pay careful attention to the explanation. Note also that the ability to perform matrix operations depends on the matrices involved being compatible in size, or dimensions, for that operation. The definition of compatible dimensions is different for different operations, so note the requirements carefully for each.

Matrix Addition and Subtraction

If two matrices have the same size, they can be added or subtracted. The operations are performed on corresponding entries.

Example \(\PageIndex{2}\)

Given the matrices \(A\), \(B\), \(C\) and \(D\), below

\[A=\left[\begin{array}{lll} 1 & 2 & 4 \\ 2 & 3 & 1 \\ 5 & 0 & 3 \end{array}\right] \quad B=\left[\begin{array}{lll} 2 & -1 & 3 \\ 2 & 4 & 2 \\ 3 & 6 & 1 \end{array}\right] \quad C=\left[\begin{array}{l} 4 \\ 2 \\ 3 \end{array}\right] \quad D=\left[\begin{array}{r} -2 \\ -3 \\ 4 \end{array}\right] \nonumber \]

Find, if possible.

As we mentioned earlier, matrix addition and subtraction involves performing these operations entry by entry.

a) We add each element of \(A\) to the corresponding entry of \(B\).

\[A+B=\left[\begin{array}{lll} 3 & 1 & 7 \\ 4 & 7 & 3 \\ 8 & 6 & 4 \end{array}\right] \nonumber \]

b) Just like the problem above, we perform the subtraction entry by entry.

\[\mathrm{C}-\mathrm{D}=\left[\begin{array}{c} 6 \\ 5 \\ -1 \end{array}\right] \nonumber \]

c) The sum \(A + D\) cannot be found because the two matrices have different sizes.

Note: Two matrices can only be added or subtracted if they have the same dimension.

Multiplying a Matrix by a Scalar

If a matrix is multiplied by a scalar, each entry is multiplied by that scalar. We can consider scalar multiplication as multiplying a number and a matrix to obtain a new matrix as the product.

Example \(\PageIndex{3}\)

Given the matrix \(A\) and \(C\) in the example above, find \(2A\) and \(- 3C\).

To find \(2A\), we multiply each entry of matrix \(A\) by 2, and to find \(-3C\), we multiply each entry of C by -3. The results are given below.

a) We multiply each entry of A by 2.

\[2 \mathrm{A}=\left[\begin{array}{ccc} 2 & 4 & 8 \\ 4 & 6 & 2 \\ 10 & 0 & 6 \end{array}\right] \nonumber \]

b) We multiply each entry of C by -3.

\[-3 C=\left[\begin{array}{c} -12 \\ -6 \\ -9 \end{array}\right] \nonumber \]

Multiplication of Two Matrices

To multiply a matrix by another is not as easy as the addition, subtraction, or scalar multiplication of matrices. Because of its wide use in application problems, it is important that we learn it well. Therefore, we will try to learn the process in a step by step manner. We first begin by finding a product of a row matrix and a column matrix.

Example \(\PageIndex{4}\)

Find the product \(AB\), given

\[A=\left[\begin{array}{lll} 2 & 3 & 4 \end{array}\right] \nonumber \]

\[B=\left[\begin{array}{l} a \\ b \\ c \end{array}\right]. \nonumber \]

The product is a \(1 \times 1\) matrix whose entry is obtained by multiplying the corresponding entries and then forming the sum.

\[\begin{align*} \mathrm{AB} &=\left[\begin{array}{lll} 2 & 3 & 4 \end{array}\right]\left[\begin{array}{l} \mathrm{a} \\ \mathrm{b} \\ \mathrm{c} \end{array}\right] \\[4pt] &=[2 (\mathrm{a}+3 \mathrm{b}+4 \mathrm{c})] \end{align*} \nonumber \]

Note that \(AB\) is a \(1 \times 1\) matrix, and its only entry is \(2a + 3b + 4c\).

Example \(\PageIndex{5}\)

\[B=\left[\begin{array}{l} 5 \\ 6 \\ 7 \end{array}\right] \nonumber \]

Again, we multiply the corresponding entries and add.

\[\begin{align*} \mathrm{AB} &=\left[\begin{array}{lll} 2 & 3 & 4 \end{array}\right]\left[\begin{array}{l} 5 \\ 6 \\ 7 \end{array}\right] \\[4pt] &=[2 \cdot 5+3 \cdot 6+4 \cdot 7]\\[4pt] &=[10+18+28]\\[4pt] &=[56] \end{align*} \nonumber \]

Note: In order for a product of a row matrix and a column matrix to exist, the number of entries in the row matrix must be the same as the number of entries in the column matrix.

Example \(\PageIndex{6}\)

Find the product AB, given

\[B=\left[\begin{array}{ll} 5 & 3 \\ 6 & 4 \\ 7 & 5 \end{array}\right]. \nonumber \]

We know how to multiply a row matrix by a column matrix. To find the product \(AB\), in this example, we will multiply the row matrix \(A\) to both the first and second columns of matrix \(B\), resulting in a \(1 \times 2\) matrix.

\[\mathrm{AB}=\left[\begin{array}{lll} 2 \cdot 5+3 \cdot 6+4 \cdot 7 & 2 \cdot 3+3 \cdot 4+4 \cdot 5 \end{array}\right]=\left[\begin{array}{ll} 56 & 38 \end{array}\right] \nonumber \]

We multiplied a \(1 \times 3\) matrix by a matrix whose size is \(3 \times 2\). So unlike addition and subtraction, it is possible to multiply two matrices with different dimensions, if the number of entries in the rows of the first matrix is the same as the number of entries in the columns of the second matrix.

Example \(\PageIndex{7}\)

Find the product \(AB\), given:

\[A=\left[\begin{array}{lll} 2 & 3 & 4 \\ 1 & 2 & 3 \end{array}\right] \nonumber \]

\[\mathrm{B}=\left[\begin{array}{ll} 5 & 3 \\ 6 & 4 \\ 7 & 5 \end{array}\right] \nonumber \]

This time we are multiplying two rows of the matrix \(A\) with two columns of the matrix \(B\). Since the number of entries in each row of \(A\) is the same as the number of entries in each column of \(B\), the product is possible. We do exactly what we did in the last example. The only difference is that the matrix \(A\) has one more row.

We multiply the first row of the matrix \(A\) with the two columns of \(B\), one at a time, and then repeat the process with the second row of A. We get

\[\mathrm{AB}=\left[\begin{array}{lll} 2 & 3 & 4 \\ 1 & 2 & 3 \end{array}\right]\left[\begin{array}{ll} 5 & 3 \\ 6 & 4 \\ 7 & 5 \end{array}\right]=\left[\begin{array}{ll} 2 \cdot 5+3 \cdot 6+4 \cdot 7 & 2 \cdot 3+3 \cdot 4+4 \cdot 5 \\ 1 \cdot 5+2 \cdot 6+3 \cdot 7 & 1 \cdot 3+2 \cdot 4+3 \cdot 5 \end{array}\right] \nonumber \]

\[\mathrm{AB}=\left[\begin{array}{ll} 56 & 38 \\ 38 & 26 \end{array}\right] \nonumber \]

Example \(\PageIndex{8}\)

Find, if possible:

\[\mathrm{E}=\left[\begin{array}{ll} 1 & 2 \\ 4 & 2 \\ 3 & 1 \end{array}\right] \quad \mathrm{F}=\left[\begin{array}{ll} 2 & -1 \\ 3 & 2 \end{array}\right] \quad \mathrm{G}=\left[\begin{array}{lll} 4 & 1 \end{array}\right] \quad \mathrm{H}=\left[\begin{array}{l} -3 \\ -1 \end{array}\right] \nonumber \]

a) To find \(EF\), we multiply the first row \(\left[\begin{array}{ll} 1 & 2 \end{array}\right]\)

of E with the columns \(\left[\begin{array}{l} 2 \\ 3 \end{array}\right] \text { and }\left[\begin{array}{l} 1 \\ -2 \end{array}\right]\) of the matrix F, and then repeat the process by multiplying the other two rows of E with these columns of F. The result is as follows:

\[\mathrm{EF}=\left[\begin{array}{ll} 1 & 2 \\ 4 & 2 \\ 3 & 1 \end{array}\right]\left[\begin{array}{cc} 2 & -1 \\ 3 & 2 \end{array}\right]=\left[\begin{array}{cc} 1 \cdot 2+2 \cdot 3 & 1 \cdot-1+2 \cdot 2 \\ 4 \cdot 2+2 \cdot 3 & 4 \cdot-1+2 \cdot 2 \\ 3 \cdot 2+1 \cdot 3 & 3 \cdot-1+1 \cdot 2 \end{array}\right]=\left[\begin{array}{cc} 8 & 3 \\ 14 & 0 \\ 9 & -1 \end{array}\right] \nonumber \]

b) Product \(FE\) is not possible because the matrix F has two entries in each row, while the matrix E has three entries in each column. In other words, the matrix F has two columns, while the matrix E has three rows.

c) \[\mathrm{FH}=\left[\begin{array}{cc} 2 & -1 \\ 3 & 2 \end{array}\right]\left[\begin{array}{c} -3 \\ -1 \end{array}\right]=\left[\begin{array}{c} 2 \cdot-3+-1 \cdot-1 \\ 3 \cdot-3+2 \cdot-1 \end{array}\right]=\left[\begin{array}{c} -5 \\ -11 \end{array}\right] \nonumber \]

d) \[\mathrm{GH}=\left[\begin{array}{ll} 4 & 1 \end{array}\right]\left[\begin{array}{l} -3 \\ -1 \end{array}\right]=\left[\begin{array}{l} 4 \cdot-3+1 \cdot-1 \\ -1 \end{array}\right]=[-13] \nonumber \]

e) \[\mathrm{HG}=\left[\begin{array}{l} -3 \\ -1 \end{array}\right]\left[\begin{array}{ll} 4 & 1 \end{array}\right]=\left[\begin{array}{ll} -3 \cdot 4 & -3 \cdot 1 \\ -1 \cdot 4 & -1 \cdot 1 \end{array}\right]=\left[\begin{array}{cc} -12 & -3 \\ -4 & -1 \end{array}\right] \nonumber \]

We summarize some important properties of matrix multiplication that we observed in the previous examples.

In order for product \(\bf{AB}\) to exist:

• the number of columns of \(\bf{A}\) must equal the number of rows of \(\bf{B}\)
• if matrix \(\bf{A}\) has dimension \(\bf{m \times n}\) and matrix \(\bf{B}\) has dimension \(\bf{n \times p}\) , then the product \(\bf{AB}\) will be a matrix with dimension \(\bf{m \times p}\) .

Matrix multiplication is not commutative: if both matrix products \(\bf{AB}\) and \(\bf{BA}\) exist, most of the time \(\bf{AB}\) will not equal \(\bf{BA}\) .

Example \(\PageIndex{9}\)

Given matrices \(R\), \(S\), and \(T\) below, find \(2RS - 3ST\).

\[R=\left[\begin{array}{lll} 1 & 0 & 2 \\ 2 & 1 & 5 \\ 2 & 3 & 1 \end{array}\right] \quad S=\left[\begin{array}{lll} 0 & -1 & 2 \\ 3 & 1 & 0 \\ 4 & 2 & 1 \end{array}\right] \quad T=\left[\begin{array}{lll} -2 & 3 & 0 \\ -3 & 2 & 2 \\ -1 & 1 & 0 \end{array}\right] \nonumber \]

We multiply the matrices R and S.

\begin{aligned} &\mathrm{RS}=\left[\begin{array}{ccc} 8 & 3 & 4 \\ 23 & 9 & 9 \\ 13 & 3 & 5 \end{array}\right]\\ &\begin{array}{l} 2 \mathrm{RS}=2\left[\begin{array}{ccc} 8 & 3 & 4 \\ 23 & 9 & 9 \\ 13 & 3 & 5 \end{array}\right]=\left[\begin{array}{ccc} 16 & 6 & 8 \\ 46 & 18 & 18 \\ 26 & 6 & 10 \end{array}\right] \\ \mathrm{ST}=\left[\begin{array}{ccc} 1 & 0 & -2 \\ -9 & 11 & 2 \\ -15 & 17 & 4 \end{array}\right] \\ 3 \mathrm{ST}=3\left[\begin{array}{ccc} 1 & 0 & -2 \\ -9 & 11 & 2 \\ -15 & 17 & 4 \end{array}\right]=\left[\begin{array}{ccc} 3 & 0 & -6 \\ -27 & 33 & 6 \\ -45 & 51 & 12 \end{array}\right] \end{array} \end{aligned}

\[2 \mathrm{RS}-3 \mathrm{ST}=\left[\begin{array}{ccc} 16 & 6 & 8 \\ 46 & 18 & 18 \\ 26 & 6 & 10 \end{array}\right]-\left[\begin{array}{ccc} 3 & 0 & -6 \\ -27 & 33 & 6 \\ -45 & 51 & 12 \end{array}\right]=\left[\begin{array}{ccc} 13 & 6 & 14 \\ 73 & -15 & 12 \\ 71 & -45 & -2 \end{array}\right] \nonumber \]

Example \(\PageIndex{10}\)

Find \(F^2\) given matrix

\[\mathrm{F}=\left[\begin{array}{ll} 2 & -1 \\ 3 & 2 \end{array}\right] \nonumber \]

\(F^2\) is found by multiplying matrix \(F\) by itself, using matrix multiplication.

\[\mathrm{F}^{2}=\left[\begin{array}{cc} 2 & -1 \\ 3 & 2 \end{array}\right]\left[\begin{array}{cc} 2 & -1 \\ 3 & 2 \end{array}\right]=\left[\begin{array}{cc} 2 \cdot 2+(-1) \cdot 3 & 2 \cdot(-1)+(-1) \cdot 2 \\ 3 \cdot 2+2 \cdot 3 & 3 \cdot(-1)+2 \cdot 2 \end{array}\right]=\left[\begin{array}{cc} 1 & -4 \\ 12 & 1 \end{array}\right] \nonumber \]

Note that \(F^2\) is not found by squaring each entry of matrix \(F\). The process of raising a matrix to a power, such as finding \(F^2\), is only possible if the matrix is a square matrix.

USING MATRICES TO REPRESENT A SYSTEM OF LINEAR EQUATIONS

In this chapter, we will be using matrices to solve linear systems. In section 2.4, we will be asked to express linear systems as the matrix equation \(\bf{AX = B}\), where \(A\), \(X\), and \(B\) are matrices.

• Matrix \(A\) is called the coefficient matrix.
• Matrix \(X\) is a matrix with 1 column that contains the variables.
• Matrix \(B\) is a matrix with 1 column that contains the constants.

Example \(\PageIndex{11}\)

Verify that the system of two linear equations with two unknowns:

\begin{array}{l} a x+b y=h \\ c x+d y=k \end{array}

can be written as \(AX = B\), where

\[A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \quad X=\left[\begin{array}{l} x \\ y \end{array}\right] \text { and } B=\left[\begin{array}{l} h \\ k \end{array}\right] \nonumber \]

If we multiply the matrices \(A\) and \(X\), we get

\[A X=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} a x+b y \\ c x+d y \end{array}\right] \nonumber \]

If \(AX = B\) then

\[\left[\begin{array}{l} a x+b y \\ c x+d y \end{array}\right]=\left[\begin{array}{l} h \\ k \end{array}\right] \nonumber \]

If two matrices are equal, then their corresponding entries are equal. It follows that

Example \(\PageIndex{12}\)

Express the following system as a matrix equation in the form \(AX = B\).

\begin{array}{l} 2 x+3 y-4 z=5 \\ 3 x+4 y-5 z=6 \\ 5 x \quad-6 z=7 \end{array}

This system of equations can be expressed in the form \(AX = B\) as shown below.

\[\left[\begin{array}{ccc} 2 & 3 & -4 \\ 3 & 4 & -5 \\ 5 & 0 & -6 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 5 \\ 6 \\ 7 \end{array}\right] \nonumber \]

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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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How to Make a Responsibility Assignment Matrix for a Project (Template Included)

The most important resource you’ll employ to deliver the project is people. They have to fit into the schedule and maintain the project budget. Defining what their roles and responsibilities are when executing tasks and delivering on the project goals is an important part of controlling the project.

How can you coordinate all the people who are involved in a project so they know what they’re doing and don’t block others from doing what they are assigned? Using a responsibility assignment matrix can help. An assignment matrix gives your project a team that gets things done.

What is a Responsibility Assignment Matrix in Project Management?

A responsibility assignment matrix (RAM) is a project management chart used to identify and define the various people and organizations and outline each of their roles in working on tasks or delivering a part of the project.

Project managers use an assignment matrix to clarify what cross-functional teams do within the boundaries of the project and its numerous processes. Sometimes a responsibility assignment matrix is required when responding to a request for proposal (RFP).

The responsibility assignment matrix can also be called a RACI matrix, which stands for responsible, accountable, consulted and informed.

• Responsible: Notes who is responsible for executing the task, which is then assigned to them.
• Accountable: Notes who has decision-making authority and how that power is delegated throughout the project team.
• Consulted: Notes who is able to offer insight into the task, from team members to stakeholders.
• Informed: Notes who is updated on what in terms of progress and performance, as well as when and how this information is disseminated.

This creates a map of connections between activities and project team members. Depending on the size of the project, there can be several assignment matrices used for various project levels.

Why Create a Responsibility Assignment Matrix?

The assignment matrix identifies what everyone on the team is responsible for, which means not only what their duties are, but how they participate in the project. Some will have defined tasks, others will offer help with work, while there are some who are designated as decision-makers. These groups all have an identity and function within the project to help guide it towards a successful end.

Clear communication leads to more efficient projects. An assignment matrix facilitates better communication between team members and provides transparency by creating a system to make sure everyone is updated and always on the same page. Belaboring communications can bog down a project with too many pointless meetings and confusing interactions in which people try to understand what they’re supposed to be doing. Using the responsibility assignment matrix helps, but having project management software that connects teams in real-time is ideal.

ProjectManager manages project information by allowing teams to attach files directly to tasks, and our unlimited file storage keeps important project documents at your fingertips anywhere, anytime. Commenting on tasks can save time and tagging others in the project team creates a communication process that avoids the pitfalls of redundancies or unnecessary meetings.

When Should a Responsibility Assignment Matrix Be Created?

The responsibility assignment matrix would be created at the start of the project. You’d want to have everyone on the project team aware of where they stand in terms of their involvement before they start executing tasks.

As much as its use is a preventative measure, it can be used prescriptively. If you’re deep into the project and things are not moving as planned, there could be communication gridlock. If team members are not in the loop, or misconstrue what they’re supposed to be doing, using a responsibility assignment matrix might untie up those knots in the communication channel.

If there’s a problem with leadership overruling suggestions on how to advance the project and this is seen as a problem, it’s likely that the roles and responsibilities of the project team need refining. The responsibility assignment matrix defines who has authority to make decisions and using it or revisiting can determine if the right people are in that position.

In fact, any of the definitions might need reexamining at any phase in the project. Perhaps tasks are falling behind schedule. This could be because team members aren’t aware of what tasks they own. Anytime a delay occurs, returning to the assignment matrix is a good first step, even if you went through the process as you should during the planning stage of the project.

How to Create a Responsibility Assignment Matrix

The actual making of a responsibility assignment matrix is not as difficult as getting everyone on board with what their roles and responsibilities are.

Therefore, you want to include your team in the process, get their input and eventually buy-in without spending too much time and energy on the process. Follow these steps to make sure everyone is in agreement and you’ll have a successful responsibility assignment.

• Identify all the participants involved in the project, from team members to stakeholders and everyone in between.
• List all deliverables associated with the project. Use a work breakdown structure to make sure you don’t miss any.
• Meet with team members on how to execute the tasks to create the deliverables. Every task needs to be discussed in terms of the team’s responsibility and authority.
• Draft the responsibility assignment matrix using a table with the project tasks listed on the left-hand column. Across the top add the name of everyone in the project.
• Where the tasks meet the project team member, assign whether they’re responsible, accountable, consulted or informed.
• When completed, share the responsibility assignment matrix with the project team and stakeholders and hold a meeting if necessary to make sure everyone understands their part in the project. If you’re working in a shared space, print out a copy and post it.

Free Responsibility Assignment Matrix Template

Using a RACI template is a shortcut that sets up your team and the project for success. ProjectManager is more than an award-winning software that organizes tasks, teams and projects to streamline work and boost productivity, it’s also the online hub for all things project management.

Among the hundreds of blog posts, guidebooks and tutorial videos are dozens of free templates that can help you through every phase of your project’s life cycle. Using our free RACI template will help you guide all the project teams better, allowing them to know where they stand in relation to the project and what their level of responsibility and accountability is.

Use it at the start of the project to avoid delays and untangle any communicative knots that are preventing the project from progressing as planned. To keep your project on track, download our free RACI template and get a head start on building a workable responsibility assignment matrix.

Best Practices

Using our free RACI template is a good start, but you have to make sure you fill it in correctly. A responsibility assignment matrix is only as good as the effort put into creating it. Here are some best practices to apply when you’re in the process of building your assignment matrix.

• Involve the team: They’re the ones who will be executing the work. You want their input and buy-in to avoid any costly mistakes or time-consuming questions about what wasn’t made clear at the beginning of the project.
• Identify every single task: Identify all the tasks required to reach your final deliverable. Once you have that thorough list make sure that there is only one person on the team who is accountable.
• Update your RACI regularly: Make sure that each new one is clearly marked as the most current version and is distributed to everyone on the team. There will be times when you’ll want to revisit the responsibility assignment matrix or changes in personnel will require an edit.
• Share responsibility viably: One person shouldn’t have to shoulder the bulk of the responsibilities for the project and you want to give authority throughout the project team and not just among the very top management team.
• Optimize tasks: Managers can use the RACI matrix to see if too many team members have been assigned to a task. Maybe these workers could be spread out for greater productivity. There could be too many people listed as consulted, which slows down the process. The assignment matrix is endlessly useful.

How ProjectManager Helps You Manage Projects Better

ProjectManager is a cloud-based tool that connects everyone in real-time to facilitate planning, monitoring and reporting on the project. It works to give everyone on the project team a job and the knowledge as to where they have authority and when to consult others, as well as defining the reporting process.

Let’s look at the people who are responsible, for example, the team who execute the project. Once invited into the software, you can share the project plan, assign them tasks, add detailed direction, add a deadline and tag for priority and more. The teams can then collaborate by attaching files and images to the tasks and commenting in real-time to work better together.

Those who need to stay informed of the project can do so by also getting invited into the project and sharing plans and schedules with them. Stakeholders can stay updated with reporting features that can generate reports on project variance, cost, time and more with one click. Then share them as a PDF. Reports can even be quickly filtered to zero in on the data stakeholders are interested in.

The responsibility assignment matrix can help you reallocate your resources when things aren’t progressing as planned. Use our software to get further insight. The resource management features include a workload chart that’s color-coded so it’s easy to see who has too many tasks and who can take on more work. Then you can simply reallocate those resources from the workload page to help your team work more productively.

ProjectManager gets you organized, keeps your team focused on their tasks and stakeholders in the loop. Gain efficiencies throughout every aspect of your project’s life cycle with an online Gantt chart to schedule work and kanban boards, a visual workflow feature that provides transparency into production. All that and it’s on a collaborative platform to keep everyone connected. Try ProjectManager today for free.

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Responsibility Assignment Matrix: A Complete Overview

Dive into the world of Responsibility Assignment Matrix (RAMs), which helps assign roles and responsibilities and how they streamline Project Management. This comprehensive blog explains their purpose, benefits, and practical use, enabling effective role definition and accountability in project teams. Continue reading!

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This comprehensive blog aims to provide you with a complete overview of the Responsibility Assignment Matrix and its pivotal role in Project Management and organisational structure. 

Table of Content 

1) What is a Responsibility Assignment Matrix in Project Management? 

2)  Responsibility Assignment Matrix (RAM) goal in Project Management 

3) How to create a Responsibility Assignment Matrix? 

4) Benefits of Responsibility Assignment Matrix 

5) Developing Responsibility Assignment Matrix (RAM) best practices 

6) Conclusion  

What is a Responsibility Assignment Matrix in Project Management? 

A Responsibility Assignment Matrix (RAM) in project management is a tool that outlines and defines the roles and responsibilities of individuals or groups involved in a project. Its purpose is to ensure that everyone understands their specific duties and tasks. The primary purpose of a RAM is to bring clarity to the project structure, helping to prevent confusion, overlap, and accountability issues throughout the project lifecycle.

RAM in Project Management is also known as Responsible, Accountable, Consulted and Informed (RACI). RACI represents different levels of roles and responsibilities for individuals or teams:

a) Responsible: The individual or group in responsible for finishing a certain job or project. They are the ones who perform the work.

b) Accountable: The individuals who have complete responsibility and decision-making authority over the job or result. They ensure that the task is completed and of satisfactory quality.

c) Consulted: Individuals or stakeholders are consulted for their views or skills before to making a decision or taking actions. They contribute essential insights but may not be directly responsible for the task.

d) Informed: Individuals or stakeholders who need regular updates on the task’s progress or result. They are not actively involved in its conclusion, but they must be notified of any advancements.  

Responsibility Assignment Matrix goal in Project Management 

The primary goal of a Responsibility Assignment Matrix in Project Management is to clearly define and communicate the roles and responsibilities of individuals or teams involved in a project. Here are the key goals of using a RAM in Project Management: 

a) Clear roles and responsibilities: The RAM establishes clear roles and responsibilities for each team member, minimising confusion and conflicts.

b) Enhanced communication: Documenting roles and responsibilities concisely in the RAM facilitates effective communication within the project team. Also, enabling quick identification of contacts for specific issues or inquires.

c) Conflict resolution: BY operating a reference point, the RAM helps to resolve conflicts or misunderstandings about responsibilities, providing a foundation for conversation and conflict resolution.

d) Improved project control: With the RAM in place, Project Managers and stakeholders can more easily monitor project progress, identifying task accountability and monitoring work package status.

e) Efficiency and accountability: By allocating responsibility to each project aspect, the RAM promotes accountability among team members, resulting in increased efficiency as everyone understands their duties and expectations.

f) Risk Management: The clear roles of RAM help detect potential hazards associated with functional gaps or overlaps, enabling proactive risk mitigation strategies.  

g) Optimal Resource Allocation: Project managers can optimise resource allocation by understanding task ownership and workload distribution as described in the RAM.

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How to create a Responsibility Assignment Matrix? 

Identify the project scope 

Start by recognising the project's scope or the exact purpose for which you are building RAM. This includes establishing project objectives, outcomes, and critical milestones.

Define roles and responsibilities 

a) Identify the key roles involved in the project. Common roles may include Project Manager, Team Leader, Team Member, Stakeholders, and other relevant positions. 

b) Clearly define the responsibilities associated with each role. These responsibilities should be specific and measurable so there is no ambiguity. 

Choose a framework - RACI, RASCI, or DACI 

a) Select a framework for your RAM. The most commonly used frameworks are RACI, RASCI, and DACI: 

b) RACI: RACI   Stands for Responsible, Accountable, Consulted, and Informed. It defines who is responsible for a task, who is accountable for its completion, who needs to be consulted, and who needs to be informed. 

c) RASCI: RASCI Similar to RACI, but with an additional role, the "S" for Support. This framework further clarifies who provides support for a task. 

d) DACI: This framework is similar to RASCI but adds the role of Driver. The Driver is responsible for ensuring that a task is completed. 

Assign roles using the framework 

a) For each task or work package within the project, assign the relevant roles using the chosen framework. Each task should have a Responsible person, an Accountable person, and, if necessary, people who need to be Consulted, Informed, or Supported. 

b) Be specific and ensure that there is only one person designated as "Accountable" for each task to avoid confusion. 

Document the RAM 

a) Create a table or chart that lists all the tasks or work packages on one axis and the identified roles on the other. 

b) Fill in the matrix with the appropriate role designations (R, A, C, I, S, D) for each task and role. 

Review and validate 

Share the RAM with the project team and stakeholders for evaluation and approval. Confirm that everyone agrees on the roles and responsibilities.

Implement and communicate 

Once the RAM has been completed and approved, communicate it to the project team and other stakeholders. Ensure that everyone understands their jobs and responsibilities. 

Regularly update the RAM 

As the project evolves, it is critical to improve the RAM as necessary. Roles and duties might shift, and new tasks may develop. Keep the RAM current to reflect the project's evolving needs.

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Benefits of Responsibility Assignment Matrix

a) Clarify roles and responsibilities: A RAM clearly defines who is responsible for each task, which helps prevent confusion and ensures that team members understand their roles.

b) Improved communication: The RAM serves as a central reference point for roles and responsibilities, promoting effective communication within the project team. Team members can quickly identify who to contact for specific issues or questions.

c) Conflict resolution: When there is a dispute or ambiguity regarding responsibilities, the RAM provides a basis for discussion and conflict resolution. It helps identify where accountability lies and facilitates problem-solving.

d) Enhanced project control: The RAM allows Project Managers and stakeholders to monitor and manage the project's development.  It helps you to track the status of assignments and ensure that tasks are getting done as planned.

e) Efficiency and accountability: Clearly defined roles and responsibilities create a sense of responsibility among team members, maybe resulting in improved productivity. When people understand what is required of them, and they are more likely to fulfil it.

f) Resource allocation: The RAM helps Project Managers optimise resource allocation by understanding who is responsible for specific tasks. This ensures that workloads are distributed evenly and that resources are used efficiently. 

Developing Responsibility Assignment Matrix (RAM) best practices 

Developing a Responsibility Assignment Matrix (RAM) is a critical aspect of project management. To create an effective RAM, consider the following best practices: 

a) Involve key stakeholders: Define roles and responsibilities after consulting with project stakeholders, team members, and subject matter experts. Their participation can provide useful information about the project's objectives. 

b) Keep it simple: Use a simple framework (e.g., RACI, RASCI, or DACI) that team members are able to quickly understand and use.

c) Use clear and specific language: In order to prevent confusion, write tasks clearly and precisely. Avoid using confusing or unclear terms that may lead to confusion.

d) Designate a single "accountable" person: Assign only one person as "Accountable" for each task. This individual is ultimately responsible for the task's completion. Multiple accountable persons can lead to confusion and accountability issues. 

e) Consult and inform appropriately:   It is important to carefully consider the individuals who need to be consulted and informed for each task. Avoid unnecessary involvement, which can lead to inefficiency. Ensure that the right people are included in these roles. 

f) Review and validate with the team: Share the RAM with the project team and stakeholders for feedback and validation. Ensure that all parties agree with the assigned roles and responsibilities. 

g) Document assumptions and clarifications: If certain roles and responsibilities are based on assumptions or require clarification, document these notes alongside the RAM. This can help avoid misunderstandings in the future. 

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Conclusion  

A Responsibility Assignment Matrix is an important tool in Project Management and organisational systems. Its importance comes from its capacity to define, assign, and explain the roles and responsibilities, resulting in improved project effectiveness and performance.

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Frequently Asked Questions

Creating a Responsibility Matrix is appropriate for the project's initiation stage. It outlines roles, duties, and communication channels to promote responsibility throughout the project's lifecycle. 

The Responsibility Assignment Matrix or RACI model divides tasks as Responsible, Accountable, Consulted, or Informed. It promotes transparency and accountability among the project group.

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The responsibility assignment matrix (ram).

by Humphreys & Associates on November 26, 2013 last modified November 7, 2017

The RAM shows the WBS on one axis and the Organizational Breakdown Structure (OBS) on the other . The intersection of these axes designates who is responsible for what products or services and is known as the control account.  After detail planning is completed, the RAM is dollarized and then depicts the total budget for each CWBS element and the total budget for each respective control account manager .

The Responsibility Assignment Matrix not only helps ensure that someone is responsible for all contractual work scope ; it also verifies that there is only one individual assigned with responsibility for that work.  When developing a RAM, the level of the organization and the CWBS level should not be driven so low that costs to maintain the EVMS become excessive. Finding the most optimum balance of cost, schedule, and technical visibility generally provides the best definition for the control account.

The Responsibility Assignment Matrix is always a part of the data call information requested for EVMS reviews.  It is a good document to easily determine the value of control accounts as well as view them by organization/function and Contract Work Breakdown Structure (CWBS) element.

Traceability between the RAM and other program documentation must always be maintained .  The RAM is updated whenever there is a change to control account budgets.  The budget for a control account shown on the RAM should always equal the budget on the work authorization document for that control account.   The sum of all budgets identified to control accounts on the RAM should also equal the amount of distributed budget shown in the Contract Budget Base Log.  The sum of the control account budgets plus Undistributed Budget equals the Performance Measurement Baseline ( PMB ).   Finally, these budgets will trace to the Contract Performance Report (CPR) or the Integrated Program Management Report (IPMR) in Formats 1 and 2.

If you have any questions regarding the Responsibility Assignment Matrix (RAM) feel free to contact Humphreys & Associate s or leave a question on the blog.

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