 Matrix Diagram

The Matrix Diagram is an analysis tool that facilitates the systematic analysis of the strengths of relationships between two or more sets of elements.  It consists of a table whose main rows and columns contain the elements being inter-related, with the rest of its cells containing symbols or numbers that denote the strengths of relationship between the elements.

The elements being inter-related in a matrix diagram may be in the form of information, concepts, conditions, activities, or other intangible items, as well as physical things such as people, equipment, tools, and materials.

The matrix diagram can be used in almost all types of decision making that involves several options or alternatives, or is affected by several factors.  Examples of these include: 1) equal distribution of major and minor assignments among members of a given project; 2) selection of a process, equipment, or material for a given purpose; 3) identifying the most critical factors affecting a given problem area; 4) matching of tasks to objectives, etc.

The elements belonging to the same row or column should have something in common, so that they comprise a set that represents something.  For instance, a matrix diagram that relates various reliability tests to various failure mechanisms might show in its main row industry-standard reliability tests and on its main column commonly-encountered failure mechanisms.

The strength of relationship between each reliability test and each failure mechanism may then be denoted on the cell where they intersect with a symbol or a number (say, 1-3, with 3 denoting the strongest relationship). Table 1 shows a simplified version of such a matrix diagram. This matrix diagram shows, for instance, that if one wants to check the reliability of a set of samples with respect to package cracking and ball lifting, then TCT should be the reliability test used instead of PCT or HTOL.

Table 1. A Matrix Diagram Relating Reliability Tests to Failure Mechanisms

 TCT PCT HTOL Package Cracking 3 2 1 Corrosion 1 3 1 Ball Lifting 3 2 1 Oxide Breakdown 1 1 3

There are many types of matrices: 1) the L-shaped matrix; 2) the T-shaped matrix; 3) the Y-shaped matrix; 4) the X-shaped matrix; and 5) the C-shaped matrix.  The two most commonly used matrices, however, are the L- and T-shaped matrices.  The L-shaped matrix has a main row and a main column that form an inverted 'L' to inter-relate two sets of items directly to each other, or a single set of items to itself. The matrix shown in Figure 1 is an example of an L-shaped matrix.

On the other hand, the T-shaped matrix has its main column (or main row) separated in the middle by a single main row (or single column) that appears in the middle of the matrix.  The T-matrix is used to inter-relate two sets of items (say, sets A and B) to a common third set of items (say, set C).  The items in set A will appear on the half of the main column above the main row, while those of set B will be in the half below the main row. The items of the common set C will appear on the main row.

If the half-columns of sets A and B in the T-matrix described above are bent to allow inter-relation of items of set A to those of set B, then a Y-shaped matrix results.  Placing two T-shaped matrices back-to-back, however, will result in an X-shaped matrix, which allows the inter-relation of four sets of items to each other.  Lastly, the C-shaped matrix is a 3-dimensional matrix that interrelates three sets of elements simultaneously.

To construct a matrix diagram, the following steps are usually taken:  1) define the purpose of the matrix diagram; 2) identify what sets of elements need to be included to meet the objective of the matrix diagram; 3) assemble the best team that can inter-relate all the elements of the matrix; 4) select the matrix format; 5) choose and define the relationship symbols; and 5) complete the matrix diagram.

As an example, suppose that a supervisor wants to document the assigned tasks and expertise levels of his engineers in matrix format.  Since he needs to interrelate two different sets of information (assignments and expertise levels) to a third common set (his engineers), then the T-shaped matrix is the best format for his purpose.  Also, in this case, he has all the information he needs to fill up the matrix, so no team is formed for the task.  Had a more complex matrix been required, then the right people must be called in to form the matrix.  Figure 1 shows the T-matrix for this example. Figure 1. A T-shaped Matrix Diagram Defining the Assignments and

Expertise Levels of 3 Engineers

In the first half of the T-matrix above, graphical symbols (a circle and a triangle) were used to interrelate the elements, with the circle denoting primary responsibility and the triangle denoting secondary responsibility.  The main reason for using graphical symbols in this portion is to have an immediate visual indication of the distribution of the tasks among the engineers. One glance at the table shows that the tasks were equally distributed.

In the second half of the T-matrix, numbers were used to denote the expertise levels of the engineers.  This is because there's a need to 'grade' the various expertise levels of the engineers.  Of course, symbols may also be used for this purpose, but doing so will also require an assignment of a number to each symbol used.  Lastly, using numbers in a matrix will allow mathematical processing of the data (such as summing up the values of a row or column), which can be useful in some cases.

The matrix diagram is a very versatile tool that can be used in many applications of the manufacturing industry. Engineers who become 'matrix thinkers' gain the ability to conjure up matrix diagrams whenever the need for it arises, allowing them to explore all available options systematically before making a major decision.