## Introduction to DSM

The Design Structure Matrix (DSM – also known as the dependency structure matrix, dependency source matrix, and dependency structure method) is a general method for representing and analyzing system models in a variety of application areas.  A DSM is a square matrix (i.e., it has an equal number of rows and columns) that shows relationships between elements in a system.  Since the behavior and value of many systems is largely determined by interactions between its constituent elements, DSMs have become increasingly useful and important in recent years.  Relative to other system modeling methods, a DSM has two main advantages:

• It provides a simple and concise way to represent a complex system.

• It is amenable to powerful analyses, such as clustering (to facilitate modularity) and sequencing (to minimize cost and schedule risk in processes).

The DSM is related to other square-matrix-based methods such as a dependency map, a precedence matrix, a contribution matrix, an adjacency matrix, a reachability matrix, and an N-square diagram, and also related to non-matrix-based methods such as directed graphs, systems of equations, and architecture diagrams and other dependency models.

The use of matrices in system modeling can be traced back to the 1960s, if not earlier. However, it was not until the 1990s that the methods received relatively widespread attention.

Earlier works used graphs for system modeling, although the use of graphs in managing complex structures is generally recognized. For example, consider a system that is composed of three elements (or sub-systems): element “A”, element “B”, and element “C”. DSM works under the assumption that – for the modeling purpose – the three elements completely describe the system and characterize its behavior. A graph may be developed to represent this system pictorially. The system graph is constructed by allowing a vertex/node on the graph to represent a system element and an edge joining two nodes to represent the relationship between two system elements. The directionality of influence from one element to another is captured by an arrow instead of a simple link. The resultant graph is called a directed graph or simply a digraph.

The matrix representation of a digraph (i.e. directed graph) has the following properties:

• it is binary (i.e. a matrix populated with only zeros and ones)

• it is square (i.e. a matrix with equal number of rows and columns)

• it has n rows and columns (n is the number of nodes of the digraph)

• it has k non-zero elements, where ( k is the number of edges in the digraph)

The matrix layout is as follows: the system elements names are placed down the side of the matrix as row headings and across the top as column headings in the same order. If there exists an edge from node i to node j, then the value of element ij (row i, column j) is unity (or marked with an X). Otherwise, the value of the element is zero (or left empty). In the binary matrix representation of a system, the diagonal elements of the matrix do not have any interpretation in describing the system, so they are usually either left empty or blacked out, although many find it intuitive to think of these diagonal cells as representative of the nodes themselves.

Binary matrices are useful in systems modeling because they can represent the presence or absence of a relationship between pairs of elements in a system. A major advantage of the matrix representation over the digraph is that its compactness and ability to provide a systematic mapping among system elements allows for a detailed analysis of a limited set of elements in the context of the overall structure. As such, it therefore provides a qualitative way of dependency modeling in a formal approach.

A matrix can also represent weighted dependencies. In such a case, a numerical DSM is used. Equally, an additional column can be used to represent the weight of an element.