Task-Based DSM Simulation

Note: For a fuller explanation of the DSM-Simulation, see:
Browning, T. and Eppinger, S., "Modeling Impacts of Process Architecture on Cost and Schedule Risk in Product Development", IEEE Transactions on Engineering Management, 49(4): 428-442, 2002.

We can use a DSM-based model of a process to quantify a process configuration's expected cost and duration and their variances. Cost, duration (or schedule), and variances in both are largely a function of the number of iterations required in the process execution and the scope or impact of those iterations.  Since iterations may or may not occur (depending on a variety of variables), this model treats iterations stochastically, with a probability of occurrence depending on the particular package of information triggering rework.

This model characterizes the design process as being composed of activities that depend on each other for information.  Changes in that information cause rework.  Thus, rework in one activity can cause a chain reaction through supposedly finished and in-progress activities.  Activity rework is a function of the probability of a change in inputs (data package volatility and activity sensitivity) and the impact of a change in inputs (activity sensitivity).  The model also assumes that independent activities can work concurrently.

As input, the DSM-Sim requires a DSM model and some additional data.  For each activity, three cost and duration estimates are required:  an optimistic or best case value (BCV), a most likely value (MLV), and a pessimistic or worst case value (WCV). 

Each activity also has an associated improvement curve, which represents learning, set up times, etc.  The improvement curve is given by a percentage, the percentage of the original duration required to regenerate the outputs.  (e.g., it takes 30% of the original duration to repeat the activity second and successive times.)  Table 1 provides example activity data.  In addition, for each activity interface, the model requires an assessment of the probability of a typical change in the data causing rework for an activity and the impact of that rework should it occur.  Impact values are percentages of an activity to rework.  Example probabilities and impacts are shown in the DSMs in Table 2.

Table 1:  Activity Data for Model Input

Table2:  DSM Dimensions One (Top) and Two, Showing Rework Probabilities and Rework Impacts, Respectively

Activity cost and duration are random variables, represented by a distribution of possible values.  The BCV, MLV, and WCV are used to form a rough, triangular PDF, denoted as TriPDF(BCV,MLV,WCV) for �smaller is better� (SIB) measures such as cost and duration.  For each run, the model randomly samples a single value for each activity�s cost and duration from that activity�s TriPDFs.

The model also uses three other vectors, each with a length equal to the number of activities.  First, a sequencing vector specifies the order of the activities in the DSM -- i.e., the process configuration.  In Table 2, for instance, the sequence is simply 1, 2, �, 14.  Changing the sequencing vector allows for the exploration of alternative process configurations.  Second, a work vector, W, keeps track of the amount of work remaining to be done on each activity.  Usually, each entry in this vector is set to 100% to begin each simulation run.  Third, a 'work now' vector of Boolean entries indicates if an activity is to do work at a given time during the simulation execution.  This vector is discussed below.

The simulation uses a simple, time advancing approach.  Each run consists of a series of equal time steps, ∆t, the size of which is smaller than the duration of the shortest activity.  (E.g., if activities have durations ranging from five to 50 weeks, a reasonable ∆t could be 0.5 weeks, as activity durations are rounded off to an integer number of time steps.  Smaller time steps provide greater model resolution at the expense of greater simulation execution time.)

During each time step, the model checks for the most upstream activity requiring work and any activities that can be executed concurrently.  Activities do not begin work until their needed inputs are available from completed, upstream activities.  (Dependencies on upstream activities are shown in the lower triangular portion of the DSM.)  Work is done on available activities during the time step, and their work remaining is reduced by the fraction of their duration represented by the time step.  The cumulative process cost is increased based on the cost of this work.  Whenever an activity finishes, the model checks for potential iterations (rework for upstream activities) and second-order rework resulting from iterations using the probabilities in the DSM.  If rework occurs, its amount�a percentage of the activity given in the DSM and modified by improvement curve effects�is added to the W vector.

When all activities are complete, all W vector entries equal zero.  The model converts the number of time steps required into appropriate units and outputs this as the process duration or schedule, S, for the run.  Cumulative process cost is output as C.

Each run of the simulation yields a cost and schedule outcome for a process in a given configuration.  The simulation continues to do batches of 25 runs until the means and variances of the C and S outcome distributions stabilize to within 1%.  The user can adjust the 1% stability threshold and the run batch size between stability checks.

The simulation is implemented in a spreadsheet.  Table 3 presents the important variables used in the model, and Table 4 provides an overview of the algorithm.

Table 3:  Model Variables

Table4:  Algorithm for Each Run of Model

The DSM-Sim utilizes some simplifying assumptions and makes some work policy choices.  These choices can be altered by modifying the simulation algorithm.  The algorithm currently assumes that the most upstream activity requiring work will begin first. 

Additional activities can work concurrently with this activity if they do not depend on it (or an interim activity).  With this approach, the workflow is determined by the process configuration, thereby allowing the comparison of alternative configurations.  Note that this policy has the effect of causing an activity to stop and wait if an activity upon which it depends begins rework.  Furthermore, the algorithm currently assumes that activities only produce outputs upon completion, and that rework cannot increase the original scope of an activity (i.e., no W vector entry can exceed 100%).

Using the example data given above and running the simulation provides some example outputs.  Figure 1 shows the pattern of activity execution in a Gantt chart format for a single run.  Figure 2 shows probability mass functions (PMFs) and cumulative distribution functions (CDFs) for a large set of cost and schedule outcomes (r = 475, ∆t = 2).  The mean cost outcome, E[C], is $647k with a standard deviation, s_C, of $45k.  The median cost outcome is $640k. 

As we expect, this distribution is skewed right (skewness = 1.15).  The mean duration, E[S], is 141 days with a standard deviation, s_S, of 8 days.  Median duration is 140 days, even though the schedule distribution is skewed right (skewness = 0.95).

Figure1:  Example Gantt Chart of Activity Work Over Process Duration

Figure 2:  Cost and Schedule PMFs and CDFs Output from Model