An Introduction to the DSM Method

The use of matrices in system modeling can be traced back to Warfield in the 70's and Steward in the 80's. However, it is not until the 1990s that the method received attention and wide spread. Much of the credit in its current popularity is accredited to MIT's research in the design process modeling arena.

Earlier works started with the use of graphs for system modeling. For example, consider a system that is composed of two elements (or sub-systems): element "A" and element "B". [The two elements  are assumed to 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) is a binary (i.e. a matrix populated with only zeros and ones) square (i.e. a matrix with equal number of rows and columns) matrix with m rows and columns, and n non-zero elements, where m is the number of nodes and n 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 (column i, row 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.

In a nut shell, binary matrices for system modeling are useful in systems modeling because they can represent the presence or absence of a relationship between pairs of elements of a system. A major advantage of the matrix representation over the digraph is in its compactness and  ability to provide a systematic mapping among system elements that is clear and easy to read regardless of size.

If the system is a project represented by a set of tasks to be performed, then off-diagonal marks in a single row of the DSM represent all of the tasks whose output is required to perform the task corresponding to that row. Similarly, reading down a specific column reveals which task receives information from the task corresponding to that column. Marks below the diagonal represent forward information transfer to later (i.e. downstream) tasks. This kind of mark is called forward mark or forward information link. Marks above the diagonal depict information fed back to earlier listed tasks (i.e. feedback mark) and indicate that an upstream task is dependent on a downstream task.

There are three basic building blocks for describing the relationship amongst system elements: parallel (or concurrent), sequential (or dependent) and coupled (or interdependent).

Three Configurations that Characterize a System
Relationship	Parallel	Sequential	Coupled
Graph Representation
DSM Representation	A B A      B	A B A      B X	A B A   X B X

In the parallel configuration, the system elements do not interact with each other. Understanding the behavior of the individual elements allow us to completely understand the behavior of the system. If the system is a project, then system elements would be project tasks to be performed. As such, activity B is said to be independent of activity A and no information exchange is required between the two activities.

In the sequential configuration, one element influences the behavior or decision of another element is a uni-directional fashion. That is, the design parameters of system element B are selected based on the system element A design parameters. Again, in terms of project tasks, task A has to be performed first before task B can start.

Finally, in the coupled system, the flow of influence or information is intertwined: element A influences B and element B influences A. This would occur if parameter A could not be determined (with certainty) without first knowing parameter B and B could not be determined without knowing A. This cyclic dependency is called "Circuits" or "Information Cycles".

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