Modeling Design Iteration Using Signal Flow Graphs

This section is based on the work of Eppinger, Nukala and Whitney (1997).

The signal flow graph represents a diagram of relationships among a number of variables. When these relationships are linear, the graph represents a system of simultaneous linear algebraic equations. The signal flow graph, as shown in the figure below, is composed of a network of directed branches which connect at the nodes. A branch jk, beginning at node j and terminating at node k, indicates its direction from j to k by an arrowhead on the branch. Each branch jk has associated with it a quantity known as the branch transmission P_jk .

For our modeling processes, the branches represent the tasks being worked (an activity-on-arc representation). The branch transmissions include the probability and time to execute the task represented by the branch:

P_jk = p_jk

where:     p_jk is the probability associated with the branch.

                t_jk is the time taken to traverse the branch

z is the transform variable used to connect the physical system (time domain) to the quantities used in the analysis (transform domain). The z transform simplifies the algebra , as it enables us to incorporate the quantities to be multiplied (probabilities) in the coefficient of the expression, and to include the quantities to be added (task times) in the exponent. The resulting system is then analogous to a discrete sampled data system, and the body of literature on this subject can be applied for the analysis thereof.

The path transmission is defined as the product of all branch transmissions along a single path. The graph transmission is the sum of the path transmissions of all the possible paths between two given nodes. The graph transmission is also the resulting expression on an arc connecting the two given nodes when all of the other nodes have been absorbed. In particular, we are interested in computing the graph transmission from the start to the finish nodes. Henceforth, graph transmission shall refer to the graph transmission between the start and the finish nodes, and is denoted as T_sf.

The coefficient of each term in the graph transmission is the probability associated with the path(s) it represents, and the exponent of z is the duration associated with the path(s). The graph transmission can be derived using the standard operations for the signal flow graphs. The impulse response of the graph transmission is then a function representing the probability distribution of the lead time of the process. It can be shown that the expected value of the lead time of the process is:

Numerical Example

A simple example is shown below:

The hypothetical design process is represented by the graph shown below:

The two tasks A and B (product design and tooling design) take 3 and 2 units of time respectively. Once task B is attempted, task A is reworked with probability 0.6, and once A is attempted, B is reworked with probability 0.3. Iterative repetitions of A are represented by task A'.

The graph transmission is found using the node elimination techniques. This graph transmission is given by:

The expected value of the project lead time E(L) is 7.6 units of time.

The first few terms of the probability distribution function is represented graphically in the figure below:

The distribution can be found for this simple example by performing synthetic division on T_sf to obtain the first few terms of the infinite series. The nominal (once through) time for A and B in series is 5 units of time, which occurs with probability 0.4. It is more likely (probability 0.42) that the lead time L of the process will be 8 units of time.

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